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Article

Recursive Symmetries: Chemically Induced Combinatorics of Colorings of Hyperplanes of an 8-Cube for All Irreducible Representations

School of Molecular Sciences, Arizona State University, Tempe, AZ 85287-1604, USA
Symmetry 2023, 15(5), 1031; https://doi.org/10.3390/sym15051031
Submission received: 11 April 2023 / Revised: 27 April 2023 / Accepted: 4 May 2023 / Published: 6 May 2023
(This article belongs to the Collection Feature Papers in Chemistry)

Abstract

:
We outline symmetry-based combinatorial and computational techniques to enumerate the colorings of all the hyperplanes (q = 1–8) of the 8-dimensional hypercube (8-cube) and for all 185 irreducible representations (IRs) of the 8-dimensional hyperoctahedral group, which contains 10,321,920 symmetry operations. The combinatorial techniques invoke the Möbius inversion method in conjunction with the generalized character cycle indices for all 185 IRs to obtain the generating functions for the colorings of eight kinds of hyperplanes of the 8-cube, such as vertices, edges, faces, cells, tesseracts, and hepteracts. We provide the computed tables for the colorings of all the hyperplanes of the 8-cube. We also show that the developed techniques have a number of chemical, biological, chiral, and other applications that make use of such recursive symmetries.

1. Introduction

N-dimensional hypercubes in general and the eight-dimensional hypercube (8-cube) in particular are of interest not only because they represent the potential energy surfaces of water clusters, including the octamer (H2O)8, but because they also have several novel applications in many other fields [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. Moreover, hypercubes in general provide representations of the periodic table, encompassing superheavy elements and elements that are yet to be discovered [1,2,3,4]. Hypercubes are employed in quantum similarity measures, quantum chemistry, computational chemistry, chirality, image processing, quantitative measures of shapes and stereochemistry, and so forth [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. There are several representations of hypercubes and the various hyperplanes of hypercubes. For example, the associated octeract, a representation of the 8-cube, is an isomorphic representation of the potential energy surfaces of a completely non-rigid water octamer, (H2O)8 [43], in which 256 vertices correspond to the potential minima in the potential energy surface. Consequently, the classifications of the rovibronic levels [43] of a fully nonrigid (H2O)8 require the automorphism group of the 8-cube, which is isomorphic with the eight-dimensional hyperoctahedral group or the wreath product group, S8[S2], which contains 10,321,920 permutation operations for which S8 is the permutation group of eight objects consisting of 8! operations. In a more general context, n-dimensional hypercubes, polycubes, recursive structures, and their properties have been studied over the years [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. They find numerous applications in vast areas, such as the representation theory of nonrigid molecules, genetic regulatory networks, biological modeling, finite automata, isomerization reactions, computer graphics, DNA synthetic bases, chirality, protein–protein interactions, parallel computing, visualizations, big data, and so forth [7,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29].
The double groups [44] of the wreath products are applicable to the classification of the potential energy surfaces of relativistic polymeric clusters [45,46,47,48] that could be made in supersonic expansions, for example, (PoH2)n, (LvH2)8, (FlH2)8, Sn3, gallium arsenide clusters and their heavier analogs, and so forth. Relativistic effects [45,46,47,48] make significant contributions to such molecules that contain very heavy atoms; thus, the coupling of the spin and orbital angular momenta results in double group symmetries [44]. Combinatorial enumerations of the colorings of structures or enumerations under group actions, especially those that are pertinent to hypercubes, polytopes, and color symmetries, have been the subject matter of several studies [33,34,35,36,37,38,39,40,49,50,51]. Such enumerations have several molecular, biological, and other applications to phylogentic trees, pandemic trees, intrinsically disordered proteins, genetic regulatory networks, dynamic chirality, spontaneous generation of optical activity, configuration interaction computations in quantum chemistry, and so forth [52,53,54,55,56,57,58,59,60,61,62,63,64,65].
Combinatorial enumerations of the colorings of the various hyperplanes of the 8-cube have not been considered until now. Such enumerations are extremely challenging as the generating functions must be obtained for 185 IRs for each of the eight hyperplanes of the 8-cube. The 8-cube exhibits eight types of (8-q) hyperplanes, with q ranging from 1 to 8. For example, q = 1 corresponds to 16 hepteracts, q = 2 yields 112 hexeracts…q = 7 provides 1024 edges, and q = 8 is simply 256 vertices of the 8-cube. In fact, Pólya’s theorem [20,21] of enumeration under group action becomes a special case of the combinatorial enumeration considered here when it is reduced to the totally symmetric A1 representation of the hypercube. Several other variants of Pólya’s theorem [36,37,38,39,50,51] have also been considered in the literature. Here, we have considered enumerations and generating functions for all 185 IRs and 185 conjugacy classes of the 8-cube. When the enumeration is generalized to all IRs, the results are extremely useful in a number of applications, for example, the enumeration of the nuclear spin statistics and nuclear spin species of the rovibronic levels of nonrigid molecules, multiple-quantum NMR spin functions and energy levels, the enumeration of isomerization reactions, dynamic chirality, mathematical methods pertinent to drug discovery, etc. [31,63,64,65]. Moreover, there exists no one-to-one correspondence between Pólya’s cycle types and the conjugacy classes for the hyperoctahedral groups. We therefore employ the matrix representations of the conjugacy classes in combination with the Möbius inversion technique [35] to generate the cycle types of all eight hyperplanes of the 8-cube for all 185 IRs of the wreath product group S8[S2]. Consequently, for (8-q) hyperplanes (q = 1 to 8) and for each IR of the wreath group, we obtain a generalized character cycle index (GCCI) in order to derive the combinatorial generating functions for the colorings of (n-q) hyperplanes of the 8-cube. We construct the combinatorial enumeration tables for the 4-colorings and 2-colorings of the hyperplanes of the 8-cube for all 185 IRs.

2. Recursive Symmetries, Wreath Products, and Combinatorial and Computational Techniques for the 8-Cube

Hypercubes are recursive structures, and their symmetries are also recursively defined in terms of wreath product groups. Recursivity is a topic that has been explored at multiple levels, including its philosophical implications [66]. In the present context, recursivity implies that the symmetry of the larger system is constructed from the previous levels of smaller systems in a nested manner, as defined in the ensuing sentences. We define an n-dimensional hypercube whose graph is represented by Qn recursively through the use of the cartesian product shown below:
Qn = Qn−1 × K2 for n ≥ 2 with Q1 = K2,
where K2 is a graph with two vertices connected by an edge. The adjacency matrices of hypercubes are also generated recursively using the above recursive construction. The adjacency matrix of Qn+1, AQn+1, is recursively expressible in terms of the corresponding matrices of Qn, as follows:
A Q n + 1 = A Q n I I A Q n
where I is simply an identity matrix of the order 2n × 2n.
The automorphism group of a graph is defined as a set of permutations of the vertices of the graph that preserve the adjacency matrix. Subsequently, for n-cubes, the permutations of the vertices must not break or make any edges. The automorphism group of an n-cube is simply given by the wreath product group, Sn[S2].
We restrict ourselves to particular details concerning the 8-cube and the enumeration of the colorings of eight possible hyperplanes for 185 irreducible representations of the S8[S2] group. The (8-q) hyperplanes (1 ≤ q ≤ 8) of the 8-cube are characterized by an 8 × 8 Coxeter’s configuration matrix [67]:
: 256 8 28 56 70 56 28 8 2 1024 7 21 35 35 21 7 4 4 1792 6 15 20 15 6 8 12 6 1792 5 10 10 5 16 32 24 8 1120 4 6 4 32 80 80 40 10 448 3 3 64 192 240 160 60 12 112 2 128 448 672 560 280 84 4 16
The number of (n-q) hyperplanes for an nD-hypercube is given by
N q = n q 2 q
It can be seen that the diagonal elements of the 8 × 8 configuration matrix yield the number of hyperplanes, with the first row corresponding to the vertices and the last row providing the number of hepteracts. The off-diagonal element Cij of the 8 × 8 configuration matrix provides the number of times hyperplane j occurs in hyperplane i of the 8-cube. Thus, C51 = 16 means that each penteract of the 8-cube contains 16 vertices, and so forth. Hence, the configuration matrix is fundamental to the combinatorial enumeration of the colorings of the hyperplanes of the 8-cube.
Figure 1 demonstrates the first row of the Coxeter [67] configuration matrix (q = 8) for the 8-cube, which shows 256 vertices and 1024 edges with C12 = 8, providing the degree of each vertex in Figure 1 taken from [68]. Transitive graphs, such as the one in Figure 1, have been considered by the author [69] as well as Balaban in the context of chemical isomerization graphs, such as the well-known Balaban cages [70,71]. Likewise, for the case under consideration, Figure 1 represents the isomerization rearrangements of the water octamer in the fully fluxional limit, where the breaking and remaking of all hydrogen bonds between any two water molecules can take place. This happens at a higher temperature at which facile rearrangements among the H-bonded water molecules lead to a totally nonrigid limit of the water octamer. The automorphism group of the graph in Figure 1 contains all permutations of the vertices such that no edges are made or broken, and it is provided by the wreath product group, S8[S2]. The generalized character cycle indices for all 185 IRs of the S8[S2] group and the 8 hyperplanes of the 8-cube are constructed next using the generalized matrix cycle types of the permutations.
The Möbius inversion technique [30,55] is one of the most powerful methods for obtaining a number of generating functions; in the present case, it facilitates the construction of permutational cycle types for all eight hyperplanes of the 8-cube under the action of the S8[S2] group. We accomplish this task by using the matrix generators for the conjugacy classes of the S8[S2] group in combination with the Möbius inversion method. Table 1 shows 185 different 2 × 8 matrix cycle types for the conjugacy classes of the S8[S2] group. The matrix cycle types are powerful generalizations of the Pólya cycle types used in the construction of the ordinary cycle index of a group. The matrix cycle types for the 185 conjugacy classes of the S8[S2] group are constructed using a combinatorial technique that considers the group actions of the composing groups in the wreath product S8[S2]. Let a permutation g ∈ S8 act on a set Ω of eight objects and generate a1 cycles of length 1, a2 cycles of length 2, a3 cycles of length 3, …, a8 cycles of length 8; the group action can then be represented by 1a12a23a3.…. 8a8 or a cycle type of g denoted as Tg = (a1, a2, a3, …, a8). From this structure, one obtains a 2 × 8 matrix cycle type for a conjugacy class of the wreath product in which the first row represents the action of {(g; e)} permutations for which e is the identity operation of the S2 group, with g ∈ S8, and the second row corresponds to the permutations {(g;π)} for π (12) ∈ S2 and g ∈ S8. Consequently, the matrix cycle type of any conjugacy class of the wreath product S8[S2], denoted by T(g;π), is a 2 × 8 matrix generated using the orbit structure of g ∈ Sn and the corresponding conjugacy class of S2. Thus, for (g;π), the matrix types of the conjugacy classes of S8[S2] are given by
T(g;π) = aik (1 ≤ i ≤ 2), (1 ≤ k ≤ 8),
To exemplify, take the conjugacy class {(1)(2)(3)(4)(5678);(12)} of S8[S2] (class number 38 in Table 1) given by (2)
T [ { ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5678 ) ; ( 12 ) } ] = 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
Likewise, the conjugacy class number 162 in Table 1, which corresponds to {(12345)(6)(7)(8);(12)}, is given by (3):
T [ { ( 12345 ) ( 6 ) ( 7 ) ( 8 ) ; ( 12 ) } ] = 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
Table 1 shows the results of all the matrix cycles thus obtained for all 185 conjugacy classes of S8[S2]. Moreover, Table 1 displays the order of each conjugacy class of the S8[S2] group and the cycle types of the eight hyperplanes (1 ≤ q ≤ 8) obtained using the Möbius inversion method [30,55]. The label R is assigned to each conjugacy class in Table 1 if the symmetry operation is a proper rotation of the 8-cube. This facilitates the further discrimination of colorings into chiral or achiral colorings. The order of each conjugacy class is shown in the third column of Table 1, which is readily obtained from the corresponding 2 × 8 matrix cycle type shown in Table 1. Let P(n) represent the number of partitions of an integer n, given that P(0) = 1. The order any conjugacy class of S8[S2] with the matrix type T(g;π) = aik is given by (4):
| T g ; π | = 8 ! 2 8 i , k a i k ! ( 2 k ) a i k
For example, for the conjugacy class in Equation (3) (conjugacy class 162 in Table 1), the order is given by (5):
3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 8 ! 2 8 3 ! ( 2.1 ) 3 1 ! 2.5 1 = 21,504
The above algorithm is iterated for all 185 conjugacy classes of the 8-cube to generate the orders of the conjugacy classes shown in the third column of Table 1. Each matrix type shown in Table 1 for the conjugacy class of the S8[S2] group generates a permutation upon its action on the set of hyperplanes for each q of the 8-cube. These actions are computed in a single generating function through the Möbius inversion technique, as illustrated by Lemmis [30] for a 4D hypercube. Thus, the generating functions for the cycle types of the (8-q) hyperplanes are computed as coefficients of xq in the Qp(x) polynomial obtained using the Möbius inversion method shown in (6):
Q p x = 1 p d / p μ p / d F d x
The summation is over all divisors d of p, and μ p / d is the Möbius function defined for any integer m as:
μ(m) = 1 if one of m’s prime factors is not a perfect square and m contains an even number of prime factors;
μ(m) = −1 if m satisfies the same perfect-square condition as before but m contains odd number of prime factors;
μ(m) = 0 if m has a perfect square as one of its factors. Consequently,
μ(m) = 1, −1, −1, 0, −1, 1, −1, 0, 0, 1 … for m = 1 to 10.
In Equation (7), Fd(x) is a polynomial in x obtained from the 2 × 8 matrix cycle types shown in the second column of Table 1. We consider only the nonzero columns of the matrix cycle types of the S8[S2] group. Suppose p is the period of the matrix type shown in Table 1, and let g = gcd(k; p), p’ = k/g, h = gcd(2k; p). and p” = 2k/h; then, the function Fp(x) is given by (8):
F p ( x ) = k n c 1 + 2 x p g a 1 k 1 + 2 x p h a 2 k 2 ,   i f   h   d o e s   n o t   d i v i d e   k ; F p ( x ) = k n c 1 + 2 x p g a 1 k ,   i f   h   d i v i d e   k ,
where we take the product only over nc non-zero columns of the 2 × 8 matrix cycle type for each of the 185 classes displayed in Table 1. The coefficient of xq in Qp(x) can be obtained by substituting the various Fd polynomials in the Möbius sum (7) in which ds are divisors of p. Thus, the cycle types of all eight hyperplanes of the 8-cube are obtained in a single generating function for each conjugacy class by collecting the coefficients of xq to obtain the cycle types for (8-q) hyperplanes of the 8-cube. We now illustrate this with an example, using the matrix type in Equation (2) for conjugacy class 38 in Table 1 for the S8[S2] group.
As can be seen from the matrix shown in Equation (2), the first and fourth columns of the matrix contain non-zero values; consequently, we consider only these two columns for computing the cycle types of the hyperplanes of the 8-cube. Thus, the maximum period to consider is 8, and the possible F polynomials are therefore F8, F4, F2, and F1, as divisors of 8 are 1, 2, 4, and 8. Applying the GCD, followed by the use of Equation (10), we obtain each of these polynomials as
F1 = (1 + 2x)4
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
The Qp polynomials are obtained using the Möbius sum, (Equation (6)), as follows:
Q1 = F1 = 1 + 8x + 24x2 + 32x3 + 16x4
Q2 = 1/2 {μ(2)F1 + μ(1)F2} = 1/2 {F2 − F1} = 1/2 {(1 + 2x)4 − (1 + 2x)4} = 0
Q4 = 1/4 {μ(1)F4 + μ(2)F2 + μ(4)F1} = 1/4 {F4 − F2} = 0
Q8 = 1/8 {μ(1)F8 + μ(2)F4 + μ(4)F2 + μ(8)F1} = 1/8 {F8 − F4}
   = x + 11x2 + 52x3 + 138x4 + 224x5 + 224x6 + 128x7 + 32x8
The coefficients of the xq terms thus obtained are sorted in a tabular form shown below for all possible Qp polynomials. Once the coefficient of xq is collected for each column, we obtain the cycle type of the (8-q) hyperplanes:
Qpxx2x3x4x5x6x7x8
Q18243216
Q2
Q4
Q81115213822422412832
Cycle type1881248111328521168138822482248128832
Hyperplaneq = 1 (hepteracts)q = 2 hexeractsq = 3 penteractsq = 4 tesseractsq = 5 cubic cellsq = 6 facesq = 7 edgesq = 8 vertices
The above technique was repeated for all 185 conjugacy classes of the 8-cube, and the results are shown in columns 5–12 (Table 1) for the (8-q) hyperplanes for q = 1 through 8, respectively. The generating functions for the colorings of the the hyperplanes for each irreducible representation are obtained using the generalized character cycle index (GCCI) for the character χ of the S8 [S2] group defined by
P G χ = 1 G g ε G χ ( g ) s 1 b 1 s 2 b 2 s n b n
where the sum is taken over all permutations g ∈ G = S8 [S2] with cycle type 1b12b23b3….8b8 upon its action on the (8-q) hyperplanes of the 8-cube. The GCCIs for all 185 irreducible representations and for each of the cycle types of the (8-q) hyperplanes (Table 1) are constructed for the S8 [S2] group. Multinomial generating functions are then computed for the colorings of the (8-q) hyperplanes of the 8-cube for all irreducible representations. The generating function is defined as follows. Let [n] be an ordered partition of eight (compositions of n = 8) into p parts such that n1 ≥ 0, n2 ≥ 0, …, np ≥ 0, i = 1 p n i = n . Then, a multinomial generating function in λs, in which λs represent arbitrary weights, is computed as
λ 1 + λ 2 + + λ p n = n p n 1 n 2 n n p λ 1 n 1 λ 2 n 2 . . λ p 1 n p 1 λ p n p
where n 1 n 2 n n p are multinomials given by
n 1 n 2 n n p = n ! n 1 ! n 2 ! n p 1 ! n p !
Define D as the set of (8-q) hyperplanes which are to be colored, and let R be the set of different colors. Furthermore, let wi be a weight assigned to each color r in R. Following Pólya, the weight of a function f from D to R can then be defined as
W f = i = 1 | R | w ( f d i )
Hence, the generating functions for each of the 185 irreducible representations of the 8-cube with character χ are provided as follows:
G F χ w 1 , w 2 . . w p = P G χ { s k w 1 k + w 2 k + . + w p 1 k + w p k }
We compute the GFs for all irreducible representations of the 8-cube’s group for all 8 hyperplanes, resulting in 1480 such combinatorial GFs for the 8-cube. The coefficient of a typical term, w1n1w2n2, …, wpnp, computes the number of colorings in the set RD that transform according to the irreducible representation with character χ. Pólya’s theorem becomes a special case for one of these 185 IRs; that is, for the totally symmetric irreducible representation A1 which has the unit character values of all 185 conjugacy classes of the 8-cube. The sizes of the multinomial generators rapidly increase for such a large number of 8-cube hyperplanes, and we therefore consider only two colors in the set R for all cases except q = 1, for which 4-colorings are considered. All the computations were carried out in FORTRAN ’95, invoking the quadruple precision arithmetic, which provides over 32 digits of accuracy for the computed results.

3. Results and Discussions

Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 display our computed results for the colorings of eight different hyperplanes for q = 1–8, respectively. As there are 185 IRs, we do not show all the results for all IRs to prevent the Tables from becoming too large. Thus, we show the restricted results for the various hyperplanes. Table 2 shows the 4-colorings for the hepteracts (q = 1) for all one- and seven-dimensional irreducible representations. Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 show the binomial colorings for the hyperplanes q = 8 through q = 2 in the order of vertices (Table 3) to hexeracts (Table 9). The reason for the restriction of the binomial colorings is that as q moves away from 1, too many partitions exist as the number of hyperplanes increases both binomially and exponentially.
As can be seen from Table 2, among four one-dimensional IRs (A1–A4), only three appear in the table for the 4-colorings of the hepteracts of the 8-cube as N(A2) becomes zero for all 4-color partitions. Moreover, for the A4 representation, only the last few rows of the color partitions give rise to nonzero values. The N(A1) numbers simply yield Pólya’s equivalence classes for four colors. As can be seen from Table 2, there are two classes for the [2,14] and [3,13] color partitions. Among the 2-colorings, there are five equivalence classes for the [8,8] color partition, or there are five inequivalent ways to color the hepteracts of the 8–cube with eight black and eight white colors. Likewise, there are 138 inequivalent ways to color the hepteracts of the 8-cube with four blue colors, four red colors, four yellow colors, and four white colors (Table 2). Thus, the frequency for each color partition shown in Table 2 provides the numbers of IRs contained in the set of all the colorings of the hepteracts of the 8-cube that transform in accordance with that IR. The representations labeled A5 to A8 in Table 2 are seven-dimensional IRs of the wreath product group S8[S2]. The A6 and A7 seven-dimensional IRs appear less frequently in colorings compared to A5 and A8. This is analogous to the one-dimensional A2 and A4 IRs, which are less frequent, and A4 appears only for the highly distributed color partitions, while A2 does not appear at all for all four-colorings of the sixteen hepteracts.
Table 3 displays the 2-colorings of the vertices of the 8-cube (q = 8). As there are 256 vertices, the color partitions are of the form [n1, 256-n1]; thus, in Table 2, we have shown only n1 or, for example, the number of black colors remaining as 256-n1 whites. As all the vertices of the 8-cube are equivalent, N(A1) becomes 1 for n1 = 1. As can be seen from Table 2, there are 8 (N(A1)) inequivalent ways to color the vertices of the 8-cube with 2 black colors, 32 ways for 3 black colors, 373 ways for 4 black colors, and so forth. The maximum is reached at [128,128], which is too large to be shown accurately, even with quadruple precision. Among the results shown in Table 3, there are 1850439767413213381676300410605 inequivalent ways to color with 28 black colors. Although the A2 IR does not present for the hepteracts, all one-dimensional IRs appear for most of the binomial colorings of the vertices of the 8-cube. The only exceptions are 1–3 black colors for the A2 and A4 IRs. As the number of black colors increases, almost every one-dimensional IR appears with nearly the same frequency. This also implies that almost every coloring becomes chiral as the number of black colors increases, approaching the peak at the [128,128] color partition. There is a similar pairing of the A1 and A3 IRs, while A2 and A4 exhibit similar pairs (Table 3).
Table 4 shows the edge colorings of the 8-cube for a set of two colors. As there are 1024 edges, the number of colorings for different IRs increases quite rapidly. As can be seen from Table 4, there are 14 inequivalent ways to color with 2 black colors, 216 ways for 3 black colors, 13,143 ways to color with 4 black colors, and so forth for the edge colorings of the 8-cube. The first nonzero frequency for the A2 IR occurs for a minimum of four black colors for the A2 IR, while for A3 and A4, one would need two and four black colors, respectively. A common feature of the binomial distribution is exhibited by all edge colorings for all IRs peaking at the [512,512] color partition.
Table 5, Table 6, Table 7, Table 8 and Table 9 show the colorings of the faces, cells, tesseracts, penteracts, and hexeracts of the 8-cube for two colors. For example, there are 17 inequivalent ways to color the faces of the 8-cube with 2 black colors, while the corresponding numbers are 17, 14, 9, and 5 for the cubic cells, tesseracts, penteracts, and hexeracts, respectively. A similar pattern is exhibited by these hyperplanes for other numbers of black colors. Likewise, the similarity of the (A1, A3) and (A2, A4) pairs is shared by all hyperplanes (Table 5, Table 6, Table 7, Table 8 and Table 9). When comparing the frequencies of the A2 IR among hyperplanes, the hexeracts and hepteracts stand out, as A2 does not appear in any 2-colorings of the hepteracts, while at least six black colors are needed to produce a coloring that contains the A2 IR for the hexeracts. An important point for all hyperplanes is that there is only one equivalence class for the A1 representation when only one black color is used. Consequently, all hyperplanes of the 8-cube or any n-cube are equivalent. This follows from the highly symmetric, vertex-, edge-, and arc-transitive natures of the n-cube. Thus, the combinatorial numbers are consistent with these general features of the n-cube.

4. Chemical, Biological and other Applications of the Colorings of 8-Cube

The colorings of the hyperplanes of n-cubes have several applications in a variety of fields, such as chemistry, biology, image processing, latent symmetries in computational psychiatry, phylogenetic networks, pandemic networks, Bethe lattices, Cayley trees, and so forth. In this section, we focus on chemical and biological applications, with slight coverage of other types of applications. An obvious connection of the 8-cube is to Boolean strings of a length of 8 for which each such string is represented by the vertex of the 8-cube, giving rise to 256 Boolean strings of a length of 8. An important chemical and spectroscopic application of the 8-cube is to the water octamer, (H2O)8. Figure 1 displays the graph of the octeract with 256 vertices and 1024 edges. If every hydrogen bond of the water octmer is allowed to be broken and remade, which happens at higher temperatures, then the cluster becomes fully nonrigid. At that limit, the graph in Figure 1 would represent the isomerization graph of the water octamer. The dynamic stereochemistry, isomerization pathways, and their intimate relations to graph theory through finite topologies and borel fields were explored earlier in the context of rapid internal rotations around the C-C single bonds [52]. The isomerization graphs provide insights into reaction pathways and phenomena such as the spontaneous generation of chirality and dynamic or transient chirality. This is attributed to the existence of dl-edges or edges between chiral pairs in the dynamic isomerization graphs.
The nuclear spin statistics and combinatorics of nuclear spin functions that have a wide-range of applications, from multiple quantum NMR spectroscopy to predicting the tunneling splittings of rovibronic levels, are critical to the interpretation of the observed spectra of nonrigid molecules including water clusters and the water octamer in particular. That is, as the molecule becomes fluxional as it surmounts several potential energy barriers, its overall symmetry is described in the nonrigid molecular group, which becomes the automorphism group of the hypercube for water clusters. For fluxional complexes such as water clusters, ammonia clusters, and methane clusters, the tunneling splittings of the rovibronic levels are obtained through the use of induced representations from the irreducible representations of the rigid molecular group to the ones in the symmetry group of the nonrigid molecule. While the magnitudes of the tunneling splittings would depend on the extent of the fluxionality, the tunneling levels are predicted using the induced representations from the smaller rigid group to the larger fluxional group. Moreover, nuclear spin functions and the nuclear spin populations of the rovibronic tunneling levels are predicted from the colorings of the hyperplanes of the n-cube.
The nuclear spin functions can be envisaged as colorings from the set of the nuclei in the molecule to a set of spin colors which correspond to the various nuclear spin orientations. The possible nuclear spin orientations depend on the isotopes of the nuclei present in the molecule and the overall nuclear spin quantum number of the isotope. For example, the naturally abundant isotope of bismuth, 209Bi, exhibits a spin of 9/2 with 10 different spin orientations or 10 colors; therefore, the set of different colors would have a cardinality of 10. On the other hand, for the normal water clusters, the proton nuclei exhibit a spin of 1/2, and thus the two spin orientations of each proton of the octamer become two distinct colors; this corresponds to the 2-coloring of the hyperplanes of the 8-cube considered herein. The most stable naturally occurring isotope of oxygen, 16O, carries no nuclear spin, although another naturally occurring 17O isotope carries a nuclear spin of 5/2, giving rise to six nuclear spin orientations or colors. The 17O NMR is the technique of choice for probing the dynamics of oxygen-containing molecules. Consequently, the combinatorial numbers enumerated in Table 2 for the 4-colorings of hepteracts contain the information needed to classify the protonic nuclear spin functions of the water octamer in the nonrigid limit. That is, there are 216 proton nuclear spin functions for the water octamer, and together, they transform as reducible representations in the symmetry group of the 8-cube. The irreducible representations contained in the set of protonic nuclear spin functions are enumerated by the combinatorial numbers in Table 2 when they are extended for all 185 IRs of the 8-cube. Likewise, when the technique is extended for 6 colors and applied to all 185 IRs, we obtain the frequencies of the IRs of the nuclear spin functions for the 17O isotopes of (H217O)8. The set of such 17O nuclear functions has a cardinality of 68 for (H217O)8. Consequently, the combinatorial numbers enumerated in Tables contain rich information pertinent to a significant amount of spectroscopically important information, for example, nuclear spin species, the nuclear spin statistical weights of the rovibronic levels, and nuclear spin multiplets and thus the intensities and hyperfine structures of the rovibronic spectra of the water octamer.
In order to demonstrate the utility of the combinatorial colorings of the hyperplanes of the 8-cube, let us consider the deuterated water octamer, (D2O)8. As deuterium is a spin-1 nucleus, its bosonic spin functions can be denoted by λ, μ, and ν. Thus, there are 316 nuclear spin functions corresponding to the 3-colorings of the hepteracts of the 8-cube, and the results are contained in Table 2 for eight of the IRs. That is, all the color partitions with three or fewer parts correspond to 3-colorings, and the subsets of numbers in Table 3 therefore provide the frequencies of the corresponding IRs in the set of the 316 nuclear spin functions of (D2O)8. As yet another example, consider the 185th IR, which has a dimension of 672 in the hyperoctahedral group of the 8-cube; it shall be denoted as Γ672-2 as it is the second IR with a dimension of 672 in the character table of S8[S2]. The generating function for Γ672-2 for (D2O)8 is shown in Table 10. In Table 10, N(Γ672-2) is the frequency of Γ672-2 in the set of bosonic spin functions of the deuterium nuclei of (D2O)8 that encompass 316 nuclear spin functions. An ordered partition of [16], for example, [7 6 3], provides the spin distribution λ7μ6 ν3. From Table 10, it can be readily seen that the frequency of Γ672-2 for the color partition [7 6 3] spin distribution is 62. In this manner, the generating function obtained using the GCCI for Γ672-2 then yields all frequencies of all spin distributions. One can sort these frequencies into spin multiplets; hence, we obtain the following nuclear spin multiplets with frequencies in parentheses for the Γ672-2, as shown below:
1Γ672-2(17), 3Γ672-2(46), 5Γ672-2(64), 7Γ672-2(67), 9Γ672-2(59), 11Γ672-2(44), 13Γ672-2(28),
15Γ672-2(15), 17Γ672-2(6), 19Γ672-2(2)                       
We iterate the above process to generate the nuclear spin multiplets for each of the 185 IRs of the hyperoctahedral group of the 8-cube. Finally, we use either the Fermi–Dirac or Bose–Einstein stipulations for the overall wave function’s symmetry. In this case, as all deuterium nuclei are bosons, we stipulate that the direct product of the rovibronic wave function and the nuclear spin function must be totally symmetric, which would then give the overall nuclear spin statistical weights for each tunneling level of the (D2O)8.
We expect the 8-cube to be applicable to relativistic measures of time [10] as hypercubes serve as time representation holders. Relativistic effects are known to make significant contributions to the electronic states and spectroscopic properties of molecules that possess very heavy atoms [44,45,46]. Consequently, the incorporation of spin–orbit coupling changes the nonrigid molecular symmetry into double groups of hypercube groups. The double group symmetry corresponds to the coupling of the spin angular and orbital angular momenta; hence, two states with the same symmetry in the double group mix. Such topics can be the subject matter of future studies.
Another important application of the colorings of n-cube hyperplanes is in chirality and transient chirality. One can define an object to be chiral if it does not possess an improper axis of rotation. For this reason, we assign a symbol R to those conjugacy classes in Table 1 to designate proper rotations. The absence of R would then imply that the operation is an improper rotation. This is important in determining the chirality of the enumerated coloring. The symbol R (Table 1) is assigned for each conjugacy class by stipulating that a conjugacy class with the matrix type [aik] is a proper rotation if and only if the sum shown below is even:
k e v e n a 1 k + k o d d a 2 k
where the first sum is restricted to even columns, while the second sum is restricted to odd columns. Consequently, the developed computer code carries out the above sum for each of the 185 conjugacy classes of the 8-cube to determine if the sum is even or odd, and then the code assigns the symbol R if the sum is even to designate the operation as a proper rotation. Consequently, a coloring of the (8-q) hyperplane of the 8-cube is chiral if and only if the coloring function transforms as the chiral irreducible representation for a given ordered color partition, [n1 n2]. The chiral irreducible representation of the 8-cube is as a one-dimensional IR with +1 character value for all proper rotations or the ones that carry the label R in Table 1 and −1 for all improper rotations. The A2 irreducible representation of the S8[S2] group is chiral; hence, it can be seen from Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 that the number of chiral colorings for the (8-q) hyperplanes corresponds to the frequencies of the A2 irreducible representation in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. As can be inferred from Table 2, the A2 representation does not occur at all for the 2-colorings or for up to four colors, suggesting that there are no chiral colorings for the hepteracts of the 8-cube if one uses up to four kinds of colors (blue, yellow, red, and white). However, there are chiral colorings for some of the other hyperplane colorings, including vertex colorings and edge colorings, as can be seen from Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9.
There are several biological and biochemical applications of the colorings of n-cube hyperplanes. In particular, the 2-colorings of the vertices of the 8-cube are critical to the combinatorics of genetic networks [41], as well as the combinatorics of the colorings of phylogenic trees [59] and pandemic trees [60]. The symmetries involved in both these applications are recursive in nature, and they are expressible as wreath product groups. Each level of the phylogenetic tree involves a recursive construction relative to the previous level. On the other hand, genetic regulatory networks are important combinatorial characterizations of the evolutionary processes which are also recursive; therefore, they are represented by hypercubes [59]. Consequently, the colorings of the vertices (q = 8) of the hypercube provide information about the equivalence classes that result in considerable simplification in computing the properties that are pertinent to the genetic regulatory networks. The equivalence classes are enumerated by the totally symmetric A1 colorings of the vertices among the 185 IRs that were considered here for the 8-cube. Wallace [53,54] has illustrated several applications of hypercube symmetries for understanding the spontaneous symmetry-breaking in intrinsically disordered proteins and the dynamics of proteins in general. Furthermore, the moonlighting functions of such proteins can be better understood through the use of the recursive symmetries displayed by hypercubes. It would be interesting to explore the applications of the chiral colorings of the hyperplanes of the hypercubes, which are enumerated by the generating functions obtained herein for the chiral IR. This aspect could become the subject of future studies.
Moreover, the colorings of the 8-cube hyperplanes have applications in completely different areas, such as X-ray diffraction patterns, neutron scattering studies, quarks, magnetic symmetry, and other physics applications [72,73,74,75]. As shown by several investigators [72,73,74,75], the analysis of complex X-ray patterns, magnetic structures, their symmetries, and the neutron diffractions of materials exhibiting distortions and understanding the dynamics of different phases could be benefited by the insights derived from wreath product and other types of approaches. There are several other chemical and material applications of dynamic symmetries, for example, to O8 clusters [76,77] and polytwistane type nanomaterials [78]. Finally such combinatorial enumerations pertinent to the nuclear spin statistics of donut type polyaromatic structures [79] and holey nanographenes should be of future interest.

5. Conclusions

In the present study on recursive symmetries arising from the wreath products of the 8-cube, we have employed combinatorial and computational techniques to seek generating functions for the colorings of 8 hyperplanes of the 8-cube for all 185 IRs. These techniques combined Möbius inversion with the generalized character cycle indices and computer codes in order to enumerate the colorings of (8-q) hyperplanes (for q = 1–8). Several applications were outlined for the prediction of the tunneling splittings of rovibronic levels and their nuclear spin species and nuclear spin statistics. A few biological and material science applications were also pointed out. It is hoped that the present study will generate further interest and applications, especially in dynamic chirality, transient chirality, and the observed spontaneous generation of optical activity and other dynamic-symmetry-induced phenomena. Graph theoretical and combinatorial techniques can be especially useful in enumerating the number of phases and the number of chiral phases that are generated during phase transitions. The dynamic chirality is such an interesting phenomenon because an achiral system could separate into distinct enantiomorphic phases during phase transitions. Such applications would require the juxtaposition of the colorings of the hyperplanes of hypercubes enumerated in higher symmetries to various subgroups that would correspond to the symmetries of different phases. Such exciting topics that involve the combinatorics of hypercubes to lower symmetries could be the subject matter of future studies.

Funding

This research received no external funding.

Data Availability Statement

All data used are contained in the manuscript.

Conflicts of Interest

The author declares no conflict of interest.

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Figure 1. The octeract graph of an 8-cube, displaying the relationship among the 256 vertices of the 8-cube. The automorphism group of this graph is the 8-dimensional hyperoctahedral group or the wreath product S8[S2] comprising 10,321,920 permutations and 185 irreducible representations. (Figure reproduced from Ref. [68]).
Figure 1. The octeract graph of an 8-cube, displaying the relationship among the 256 vertices of the 8-cube. The automorphism group of this graph is the 8-dimensional hyperoctahedral group or the wreath product S8[S2] comprising 10,321,920 permutations and 185 irreducible representations. (Figure reproduced from Ref. [68]).
Symmetry 15 01031 g001
Table 1. Conjugacy Classes of the 8-dimensional hyperoctahedral group & cycle types for each hyperplane.
Table 1. Conjugacy Classes of the 8-dimensional hyperoctahedral group & cycle types for each hyperplane.
Matrix TypeOrderFd(x)q = 1q = 2q = 3q = 4q = 5q = 6q = 7q = 8
1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
R
F1 = (1 + 2x)811611121448111201179211792110241256
2 7 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8F1 = (1 + 2x)7
F2 = (1 + 2x)8
114
2
184
214
1280284156022801672256014482672112824482128
3 6 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 28
R
F1 = (1 + 2x)6
F2 = (1 + 2x)8
112
22
160
226
116021441240244011922800164286425122128
4 5 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 56F1 = (1 + 2x)5
F2 = (1 + 2x)8
110
23
140
236
180218418025201322880289625122128
5 4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 70
R
F1 = (1 + 2x)4
F2 = (1 + 2x)8
18
24
124
244
132220811625522896289625122128
6 3 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 56F1 = (1 + 2x)3
F2 = (1 + 2x)8
16
25
112
250
18222025602896289625122128
7 2 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 28
R
F1 = (1 + 2x)2
F2 = (1 + 2x)8
14
26
14
254
222425602896289625122128
8 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 8F1 = (1 + 2x)
F2 = (1 + 2x)8
12
27
256222425602896289625122128
9 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 1
R
F1 = 1
F2 = (1 + 2x)8
28256222425602896289625122128
10 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 840
R
F1 = (1 + 2x)4(1 + 2x2)2
F2 = (1 + 2x)8
1824128242164219211162502116028161160281611282448164296
11 4 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1080F1 = (1 + 2x)4(1 + 2x2)
F2 = (1 + 2x)6
F4 = (1 + 2x)8
182241262174131482564721642884220164264440013221644324256464
12 3 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x)3(1 + 2x2)2
F2 = (1 + 2x)8
1625116248132220815225341562868148287213224962128
13 3 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6720
R
F1 = (1 + 2x)3(1 + 2x2)
F2 = (1 + 2x)6
F4 = (1 + 2x)8
1623411422341312027047212421084220116288440023244324256464
14 2 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 5020
R
F1 = (1 + 2x)2(1 + 2x2)2
F2 = (1 + 2x)8
142618252116221612025501162888116288825122128
15 2 1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 10,080F1 = (1 + 2x)2(1 + 2x2)
F2 = (1 + 2x)6
F4 = (1 + 2x)8
1424416227413182764721821164220296440023244324256464
16 4 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 840
R
F1 = (1 + 2x)4
F2 = (1 + 2x)4
F4 = (1 + 2x)8
184212442213241041164276444844484256464
17 3 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x)3
F2 = (1 + 2x)4
F4 = (1 + 2x)8
162 4211226422182124104284276444844484256464
18 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 5040
R
F1 = (1 + 2x)2
F2 = (1 + 2x)4
F4 = (1 + 2x)8
1422
42
142104222164104284276444844484256464
19 1 2 0 3 0 0 0 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x)(1 + 2x2)2
F2 = (1 + 2x)8
122714254182220142558182892289625122128
20 1 1 0 3 1 0 0 0 0 0 0 0 0 0 0 0 6720
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)6
F4 = (1 + 2x)8
12254122294131427847221204220296440023244324256464
21 1 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x)
F2 = (1 + 2x)4
F4 = (1 + 2x)8
1223 422124222164104284276444844484256464
22 0 2 0 4 0 0 0 0 0 0 0 0 0 0 0 0 840
R
F1 = (1 + 2x2)2
F2 = (1 + 2x)8
281425422241425582896289625122128
23 0 1 0 4 1 0 0 0 0 0 0 0 0 0 0 0 1080F1 = (1 + 2x2)
F2 = (1 + 2x)6
F4 = (1 + 2x)8
2641222941328047221204220296440023244324256464
24 0 0 0 4 2 0 0 0 0 0 0 0 0 0 0 0 840
R
F1 = 1
F2 = (1 + 2x)4
F4 = (1 + 2x)8
24422124222164104284276444844484256464
25 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 448
R
F1 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)8
110 32140324182312211003340111235601160354411603288164364
26 4 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 2240F1 = (1 + 2x)4(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)7
F6 = (1 + 2x)8
182
32
12428
32062
134224
382620
132234
3176682
148232
32086176
164248
3128 6208
132264
3326128
232632
27 3 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 4480
R
F1 = (1 + 2x)3(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)6
F6 = (1 + 2x)8
1622
32
112214
31664
110236
350636
112244
3766132
124244
3566252
116272
3166264
2806144232632
28 2 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 4480F1 = (1 + 2x)2(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)5
F6 = (1 + 2x)8
1423
32
14218
31266
12240
326648
18246
3246158
18252
386276
28062722806144232632
29 5 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 448F1 = (1 + 2x)5
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)5
F6 = (1 + 2x)8
110614061218026611802106170132240628028062722806144232632
30 4 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 2240
R
F1 = (1 + 2x)4
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)4
F6 = (1 + 2x)8
182
6
124286121322256611162426170256628028062722806144232632
31 3 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 4480F1 = (1 + 2x)3
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)3
F6 = (1 + 2x)8
1622
6
112214612182376612506170256628028062722806144232632
32 2 0 0 3 0 1 0 0 0 0 0 0 0 0 0 0 4480
R
F1 = (1 + 2x)2
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)2
F6 = (1 + 2x)8
1423
6
142186122416612506170256628028062722806144232632
33 1 0 1 4 0 0 0 0 0 0 0 0 0 0 0 0 2240
R
F1 = (1 + 2x)(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)4
F6 = (1 + 2x)8
1224
32
22038681224031065614248
346168
256628028062722806144232632
34 1 0 0 4 0 1 0 0 0 0 0 0 0 0 0 0 2240F1 = (1 + 2x)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)
F6 = (1 + 2x)8
1224
6
2206122416612506170256628028062722806144232632
35 0 0 1 5 0 0 0 0 0 0 0 0 0 0 0 0 448F1 = (1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)3
F6 = (1 + 2x)8
25322203461012240
32660
2506170256628028062722806144232632
36 0 0 0 5 0 1 0 0 0 0 0 0 0 0 0 0 448
R
F2 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)8
F1 = 1
F3 = 1
2562206122416612506170256628028062722806144232632
37 4 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x)4(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
1842124224211322164961182494251116272440814825644081642324224132216448
38 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3360
R
F1 = (1 + 2x)4
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
1881248111328521168138822482248128832
39 3 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 13440
R
F1 = (1 + 2x)3(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
162
42
1122842118228496122574251112274440812426844081162564224232448
40 3 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 13440F1 = (1 + 2x)3
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
16
2
8
1122681118212852288138822482248128832
41 2 0 0 2 0 0 1 0 0 0 0 0 0 0 0 0 20,160F1 = (1 + 2x)2(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
1422
42
142124212324961225742511827644081827644082644224232448
42 2 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 20,160
R
F1 = (1 + 2x)2
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
1422
8
14210811216852288138822482248128832
43 1 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 13440
R
F1 = (1 + 2x)(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
1223
42
21442123249612257425114278440828044082644224232448
44 1 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 13440F1 = (1 + 2x)
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
1223
8
212811216852288138822482248128832
45 0 0 0 4 0 0 1 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
2442214421232496122574251280440828044082644224232448
46 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 3360
R
F1 = 1
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
248212811216852288138822482248128832
47 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x)(1 + 2x2)2(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)4(1 + 2x2)2
F6 = (1 + 2x)8
1224
32
14218
3868
110236
318652
18246
336652
116248
3486256
116272
3486248
18276
3406124
116224
316624
48 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 53,760F1 = (1 + 2x)(1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)4(1 + 2x2)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
1222
324
12253847
64 122
1622314418
618 1218
1424320422
628 1266
14210320422 618 12126182438436
64 12132
44012724161216
49 0 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x2)2(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x2)2(1 + 2x)3
F6 = (1 + 2x)8
253214218
34610
12240
310656
14248
3166162
18252
3166272
280316626418276
386140
232632
50 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 53,760
R
F1 = (1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x2)(1 + 2x)3
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
2332
4
1225
3447
66 122
1224
36418
622 1218
2638422
634 1266
14210
34422
626 12126
28436
6812132
44012724161216
51 1 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x)(1 + 2x2)2
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)(1 + 2x2)2
F6 = (1 + 2x)8
1224
6
142186121823766114248617018252628028062722806144232632
52 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 53,760
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)(1 + 2x2)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
1222
4
6
122547
68 122
1423418
6251218
26422
6381266
212422
62812126
28436
6812132
44012724161216
53 0 2 0 1 0 1 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x2)2
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x2)2
F6 = (1 + 2x)8
25614218612241661142486170256628028062722806144232632
54 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 53,760F1 = (1 + 2x2)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x2)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
234
6
122547
68 122
25418
6251218
26422
6381266
212422
62812126
28436
6812132
44012724161216
55 1 0 1 0 2 0 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x)(1 + 2x3)
F2 = (1 + 2x)(1 + 2x3)
F3 = (1 + 2x)4
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4
F12 = (1 + 2x)8
1232
42
3841012412310
4201228
1434
4241284
428121404401213644012724161216
56 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x3)
F2 = (1 + 2x)(1 + 2x3)
F3 = (1 + 2x)3
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4
F12 = (1 + 2x)8
2 32
42
34410
62124
1232420
641228
22424
621284
428121404401213644012724161216
57 1 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x)
F2 = (1 + 2x)(1 + 2x3)
F3 = (1 + 2x)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4
F12 = (1 + 2x)8
1242
6
410641242 420
651228
22424
621284
428121404401213644012724161216
58 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = 1
F2 = (1 + 2x)(1 + 2x3)
F3 = 1
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4
F12 = (1 + 2x)8
2 42
6
410641242 420
651228
22424
621284
428121404401213644012724161216
59 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 107,520F1 = (1 + 2x)(1 + 2x3)(1 + 2x4)
F2 = (1 + 2x)(1 + 2x3)(1 + 2x2)2
F3 = (1 + 2x)4(1 + 2x4)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4(1 + 2x2)2
F12 = (1 + 2x)8
1232
42
2238
49124
1224
310418
64 1226
162
34423
616 1276
1426
34424
622 12128
28316436
616 12124
1422
320438
610 1262
1824
38412
64 1212
60 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 107,520
R
F1 = (1 + 2x)(1 + 2x3)
F2 = (1 + 2x)(1 + 2x3)
F3 = (1 + 2x)4
F4 = (1 + 2x)(1 + 2x3)
F6 = (1 + 2x)4
F8 = (1 + 2x)5(1 + 2x3)
F12 = (1 + 2x)4
F24 = (1 + 2x)8
1232
8
388524212310
8102414
1434
8122442
81424708202468820243688248
61 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 107,520
R
F1 = (1 + 2x3)(1 + 2x4)
F2 = (1 + 2x)(1 + 2x3)(1 + 2x2)2
F3 = (1 + 2x)3(1 + 2x4)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4(1 + 2x2)2
F12 = (1 + 2x)8
2
3242
223449
62 124
1224
32418
68 1226
1223
4236181276
2834424622 121282838
436620 12124
1422
34438
618 1262
28412
681212
62 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 107,520F1 = (1 + 2x3)
F2 = (1 + 2x)(1 + 2x3)
F3 = (1 + 2x)3
F4 = (1 + 2x)(1 + 2x3)
F6 = (1 + 2x)4
F8 = (1 + 2x)5(1 + 2x3)
F12 = (1 + 2x)4
F24 = (1 + 2x)8
2
328
3462
85242
123264
8102414
2262
8122442
81424708202468820243688248
63 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 107,520
R
F1 = (1 + 2x)(1 + 2x4)
F2 = (1 + 2x)(1 + 2x3)(1 + 2x2)2
F3 = (1 + 2x)(1 + 2x4)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4(1 + 2x2)2
F12 = (1 + 2x)8
1242
6
2249
64124
25418
691226
1223
423618 1276
1426424
624 12128
28436
62412124
24438
620 1262
28412
68 1212
64 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 107,520F1 = (1 + 2x)
F2 = (1 + 2x)(1 + 2x3)
F3 = (1 + 2x)
F4 = (1 + 2x)(1 + 2x3)
F6 = (1 + 2x)4
F8 = (1 + 2x)5(1 + 2x3)
F12 = (1 + 2x)4
F24 = (1 + 2x)8
126
8
64852422 65
8102414
2262
8122442
81424708202468820243688248
65 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 107,520F1 = (1 + 2x4)
F2 = (1 + 2x)(1 + 2x3)(1 + 2x2)2
F3 = (1 + 2x4)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)4(1 + 2x2)2
F12 = (1 + 2x)8
2 42
6
2249
64124
25418
691226
1223423
6181276
28424
62412128
28436
62412124
24438
620 1262
28412
68 1212
66 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 107,520
R
F1 = 1
F2 = (1 + 2x)(1 + 2x3)
F3 = 1
F4 = (1 + 2x)(1 + 2x3)
F6 = (1 + 2x)4
F8 = (1 + 2x)5(1 + 2x3)
F12 = (1 + 2x)4
F24 = (1 + 2x)8
2 6
8
6485
242
2 65
8102414
2262
8122442
81424708202468820243688248
67 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 56F1 = (1 + 2x)6(1 + 2x2)
F2 = (1 + 2x)8
112
22
16222511842132136023801512264015442624138423201128264
68 6 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 56
R
F1 = (1 + 2x)6
F2 = (1 + 2x)6
F4 = (1 + 2x)8
11241604131160472124042201192440016444324256464
69 5 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 336
R
F1 = (1 + 2x)5(1 + 2x2)
F2 = (1 + 2x)8
110
23
1422351100217411602480119228001160281616424802128
70 5 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 336F1 = (1 + 2x)5
F2 = (1 + 2x)6
F4 = (1 + 2x)8
1102
4
1402104131802404721802804220132280440023244324256464
71 4 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 840F1 = (1 + 2x)4(1 + 2x2)
F2 = (1 + 2x)8
1824126243148220016425281642864132288025122128
72 4 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 840
R
F1 = (1 + 2x)4
F2 = (1 + 2x)6
F4 = (1 + 2x)8
1822
4
12421841313226447211621124220296440023244324256464
73 3 1 0 3 0 0 0 0 0 0 0 0 0 0 0 0 1120
R
F1 = (1 + 2x)3(1 + 2x2)
F2 = (1 + 2x)8
16
25
114249120221412425481162888289625122128
74 3 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 1120F1 = (1 + 2x)3
F2 = (1 + 2x)6
F4 = (1 + 2x)8
16
234
1122244131827647221204220296440023244324256464
75 2 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 840F1 = (1 + 2x)2(1 + 2x2)
F2 = (1 + 2x)8
14
26
162531822201825562896289625122128
76 2 0 0 4 1 0 0 0 0 0 0 0 0 0 0 0 840
R
F1 = (1 + 2x)2
F2 = (1 + 2x)6
F4 = (1 + 2x)8
14
244
1422841328047221204220296440023244324256464
77 1 1 0 5 0 0 0 0 0 0 0 0 0 0 0 0 336
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)8
12
27
1225514222225602896289625122128
78 1 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 336F1 = (1 + 2x)
F2 = (1 + 2x)6
F4 = (1 + 2x)8
12
254
23041328047221204220296440023244324256464
79 0 1 0 6 0 0 0 0 0 0 0 0 0 0 0 0 56F1 = (1 + 2x2)
F2 = (1 + 2x)8
2812255222425602896289625122128
80 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 0 56
R
F1 = 1
F2 = (1 + 2x)6
F4 = (1 + 2x)8
26423041328047221204220296440023244324256464
81 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 8960F1 = (1 + 2x)3(1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)6(1 + 2x2)
F6 = (1 + 2x)8
16
22
32
114213
31664
122230
354634
136232
31086116
144234
31566202
140260
31686188
148256
3112688
132216
332 616
82 3 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 8960
R
F1 = (1 + 2x)3(1 + 2x3)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)6
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
16
324
112316
47122
110350
4181218
112376
4221266
124356
42212126
116316
43612132
44012724161216
83 3 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8960
R
F1 = (1 + 2x)3(1 + 2x2)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)3(1 + 2x2)
F6 = (1 + 2x)8
16
226
1142136121202316611242386170116248628028062722806144232632
84 3 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 8960F1 = (1 + 2x)3
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)3
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
16
4 6
11247
68122
182418
6251218
26422
6381266
212422
62812126
28436
6812132
44012724161216
85 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x)2(1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)5(1 + 2x2)
F6 = (1 + 2x)8
14
23
32
16217
31266
110236
330646
116242
3486146
112250
3606250
116272
3486248
116272
3166136
232632
86 2 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x)2(1 + 2x3)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)5
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
142
32
4
1424
31247
62 122
1224
326418
612 1218
1822
324422
626 1266
1828
38422
624 12126
28436
6812132
44012724161216
87 2 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x)2(1 + 2x2)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)2(1 + 2x2)
F6 = (1 + 2x)8
14
236
1621761218237661182466170256628028062722806144232632
88 2 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x)2
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)2
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
142
4
6
142447
68122
25418
6251218
26422
6381266
212422
62812126
28436
6812132
44012724161216
89 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x)(1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)4(1 + 2x2)
F6 = (1 + 2x)8
12
24
32
12219
3868
16238
314654
14248
3206160
14254
3206270
18276
386268
2806144232632
90 1 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x)(1 + 2x3)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)4
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
12
22
324
263847
64122
1224
310418
6201218
1424
34422
6361266
212422
62812126
28436
6812132
44012724161216
91 1 1 0 2 0 1 0 0 0 0 0 0 0 0 0 0 26880
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x)(1 + 2x2)
F6 = (1 + 2x)8
12
246
12219612142396612506170256628028062722806144232632
92 1 0 0 2 1 1 0 0 0 0 0 0 0 0 0 0 26880F1 = (1 + 2x)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
12
224
6
2647
68122
25418
6251218
26422
6381266
212422
62812126
28436
6812132
44012724161216
93 0 1 1 3 0 0 0 0 0 0 0 0 0 0 0 0 8960
R
F1 = (1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x2)(1 + 2x)3
F6 = (1 + 2x)8
25
32
12219
34610
12240
36658
25038616614254
346278
28062722806144232632
94 0 0 1 3 1 0 0 0 0 0 0 0 0 0 0 0 8960F1 = (1 + 2x3)
F2 = (1 + 2x)3(1 + 2x3)
F3 = (1 + 2x)3
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
23
32
4
2634
4766
122
1224
32418
6241218
26422
6381266
212422
62812126
28436
6812132
44012724161216
95 0 1 0 3 0 1 0 0 0 0 0 0 0 0 0 0 8960F1 = (1 + 2x2)
F2 = (1 + 2x)5(1 + 2x3)
F3 = (1 + 2x2)
F6 = (1 + 2x)8
25
6
122196122416612506170256628028062722806144232632
96 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 0 8960
R
F1 = 1
F2 = (1 + 2x)3(1 + 2x3)
F3 = 1
F4 = (1 + 2x)5(1 + 2x3)
F6 = (1 + 2x)6
F12 = (1 + 2x)8
23
4
6
2647
68122
25418
6251218
26422
6381266
212422
62812126
28436
6812132
44012724161216
97 2 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40,320
R
F1 = (1 + 2x)2(1 + 2x2)(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
14
22
42
16211
421
18228496110253
4251
18276
4408
11227
44408
116256
4224
116224
448
98 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 40,320F1 = (1 + 2x)2(1 + 2x2)
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
14
228
1629
811
18212
852
1824
8138
822482248128832
99 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 80,640F1 = (1 + 2x)(1 + 2x2)(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
12
23
42
12213
421
14230
496
12257
4251
14278
4408
14278
4408
18260
4224
232448
100 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 80,640
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
12
23
8
12211
811
14214
852
288138822482248128832
101 2 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 40,320F1 = (1 + 2x)2(1 + 2x4)
F2 = (1 + 2x)2(1 + 2x2)2
F4 = (1 + 2x)8
14
43
1422
426
2841081229
4275
1824
4444
1824
4444
4256464
102 2 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 40,320
R
F1 = (1 + 2x)2
F2 = (1 + 2x)2
F4 = (1 + 2x)4
F8 = (1 + 2x)8
14
4
8
1445
811
48852448138822482248128832
103 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 80,640
R
F1 = (1 + 2x)(1 + 2x4)
F2 = (1 + 2x)2(1 + 2x2)2
F4 = (1 + 2x)8
12
2
43
244262841081229
4275
1426
4444
2844444256464
104 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 80,640F1 = (1 + 2x)
F2 = (1 + 2x)2
F4 = (1 + 2x)4
F8 = (1 + 2x)8
12
2
4
8
2245
811
48852448138822482248128832
105 0 1 0 2 0 0 1 0 0 0 0 0 0 0 0 0 40,320
R
F1 = (1 + 2x2)(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
24
42
12213
421
23249612257
4251
280440814278
4408
2644224232448
106 0 1 0 2 0 0 0 0 0 0 0 1 0 0 0 0 40,320F1 = (1 + 2x2)
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
24
8
12211
811
216852288138822482248128832
107 0 0 0 2 1 0 1 0 0 0 0 0 0 0 0 0 40,320F1 = (1 + 2x4)
F2 = (1 + 2x)2(1 + 2x2)2
F4 = (1 + 2x)8
22
43
244262841081229
4275
2844442844444256464
108 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 40,320
R
F1 = 1
F2 = (1 + 2x)2
F4 = (1 + 2x)4
F8 = (1 + 2x)8
22
4
8
2245
811
48852448138822482248128832
109 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40,320F1 = (1 + 2x2)2(1 + 2x4)
F2 = (1 + 2x)4(1 + 2x2)2
F4 = (1 + 2x)8
24
42
14212
421
23249616255
4251
280440818276
4408
264422418228
448
110 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 80,640
R
F1 = (1 + 2x2)(1 + 2x4)
F2 = (1 + 2x)2(1 + 2x2)2
F4 = (1 + 2x)8
22
43
1223
426
2841081229
4275
2844441426
4444
4256464
111 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 40,320
R
F1 = (1 + 2x2)2
F2 = (1 + 2x)4
F4 = (1 + 2x)4
F8 = (1 + 2x)8
24
8
14210
811
2168521426
8138
822482248128832
112 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 80,640F1 = (1 + 2x2)
F2 = (1 + 2x)2
F4 = (1 + 2x)4
F8 = (1 + 2x)8
22
4
8
122
45811
48852448138822482248128832
113 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 40,320F1 = (1 + 2x4)
F2 = (1 + 2x2)2
F4 = (1 + 2x)8
44224274112122
4279
444844484256464
114 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 40,320
R
F1 = 1
F2 = 1
F4 = (1 + 2x)4
F8 = (1 + 2x)8
4284681148852448138822482248128832
115 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1680
R
F1 = (1 + 2x2)4
F2 = (1 + 2x)8
2818252222412425482896132288025121162120
116 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6720F1 = (1 + 2x2)3
F2 = (1 + 2x)6
F4 = (1 + 2x)8
26416227
413
2804721122114
4220
296440018228
4432
4256464
117 0 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 10,080
R
F1 = (1 + 2x2)2
F2 = (1 + 2x)4
F4 = (1 + 2x)8
24
42
14210
422
21641041426
4276
444844484256464
118 0 1 0 0 3 0 0 0 0 0 0 0 0 0 0 0 6720F1 = (1 + 2x2)
F2 = (1 + 2x)2
F4 = (1 + 2x)8
22
43
122
427
41124280444844484256464
119 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1680
R
F1 = 1
F2 = 1
F4 = (1 + 2x)8
4442841124280444844484256464
120 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 17,920
R
F1 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)8
14
34
14336143148116336811635921435961163336116380
121 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 35,840F1 = (1 + 2x)2(1 + 2x3)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)5
F6 = (1 + 2x)8
14
326
14312
612
122
326661
1824
3246172
1824
386292
22629828616828640
122 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 17,920
R
F1 = (1 + 2x)2
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)2
F6 = (1 + 2x)8
14
62
146182267428618428629622629828616828640
123 1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 35,840F1 = (1 + 2x)(1 + 2x3)2
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)7
F6 = (1 + 2x)8
12
2
34
22328
64
14392
628
1824
3184692
283224
6184
143148
6224
1824
3406148
28640
124 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 71,680
R
F1 = (1 + 2x)(1 + 2x3)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)4
F6 = (1 + 2x)8
12
2
326
2238
614
122
310669
1426
346182
28629622629828616828640
125 1 0 0 1 0 2 0 0 0 0 0 0 0 0 0 0 35,840F1 = (1 + 2x)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)
F6 = (1 + 2x)8
12
2
62
226182267428618428629622629828616828640
126 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 17,920
R
F1 = (1 + 2x3)2
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)6
F6 = (1 + 2x)8
22
34
22320
68
14352
648
28380
6144
28364
6264
14320
6288
28616828640
127 0 0 1 2 0 1 0 0 0 0 0 0 0 0 0 0 35,840F1 = (1 + 2x3)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)3
F6 = (1 + 2x)8
22
326
2234
616
122
32673
28618428629622629828616828640
128 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 17,920
R
F1 = 1
F2 = (1 + 2x)2(1 + 2x3)2
F3 = 1
F6 = (1 + 2x)8
22
62
226182267428618428629622629828616828640
129 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 35,840F1 = (1 + 2x2)(1 + 2x3)2
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x2)(1 + 2x)6
F6 = (1 + 2x)8
22
34
122
32068
14360
644
283120
6124
1824
31686212
143180
6208
283128
6104
1824
340620
130 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 71,680
R
F1 = (1 + 2x2)(1 + 2x3)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x2)(1 + 2x)3
F6 = (1 + 2x)8
22
326
122
34616
122
36671
2838
6180
1426
346294
22629828616828640
131 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 35,840F1 = (1 + 2x2)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x2)
F6 = (1 + 2x)8
2262122
618
2267428618428629622629828616828640
132 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 35,840
R
F1 = (1 + 2x3)2
F2 = (1 + 2x3)2
F3 = (1 + 2x)6
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)6
F12 = (1 + 2x)8
34
4
3204
124
14352
1224
38044
1272
36444
12132
14320
12144
441284441220
133 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 71,680F1 = (1 + 2x3)
F2 = (1 + 2x3)2
F3 = (1 + 2x)3
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)6
F12 = (1 + 2x)8
32
4
6
344
68124
122
326251224
44640
1272
44632
12132
22610
12144
441284441220
134 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 35,840
R
F1 = 1
F2 = (1 + 2x3)2
F3 = 1
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)6
F12 = (1 + 2x)8
4
62
4
610124
22626
1224
44640
1272
44632
12132
22610
12144
441284441220
135 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x)2(1 + 2x2)3
F2 = (1 + 2x)8
14
26
110251124221213625421482872156286813224961322112
136 2 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 10,080
R
F1 = (1 + 2x)2(1 + 2x2)2
F2 = (1 + 2x)6
F4 = (1 + 2x)8
14
24
4
18226
413
116272
472
1202110
4220
116288
4400
116224
4432
4256464
137 2 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 10,080F1 = (1 + 2x)2(1 + 2x2)
F2 = (1 + 2x)4
F4 = (1 + 2x)8
14
22
42
1629
422
18212
4104
1824
4276
444844484256464
138 1 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6720
R
F1 = (1 + 2x)(1 + 2x2)3
F2 = (1 + 2x)8
12
27
1625311222181122554124288418289211625042128
139 1 2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 20,160F1 = (1 + 2x)(1 + 2x2)2
F2 = (1 + 2x)6
F4 = (1 + 2x)8
12
25
4
14228
413
18276
472
142118
4220
18292
4400
23244324256464
140 1 1 0 1 2 0 0 0 0 0 0 0 0 0 0 0 20,160
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)4
F4 = (1 + 2x)8
12
23
42
12211
422
14214
4104
284276444844484256464
141 2 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3360
R
F1 = (1 + 2x)2
F2 = (1 + 2x)2
F4 = (1 + 2x)8
14
43
1442741124280444844484256464
142 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 6720F1 = (1 + 2x)
F2 = (1 + 2x)2
F4 = (1 + 2x)8
12
2
43
2242741124280444844484256464
143 0 3 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3360F1 = (1 + 2x2)3
F2 = (1 + 2x)8
281625322241122554289618289225122128
144 0 2 0 2 1 0 0 0 0 0 0 0 0 0 0 0 10,080
R
F1 = (1 + 2x2)2
F2 = (1 + 2x)6
F4 = (1 + 2x)8
26
4
14228
413
280472142118
4220
296440023244324256464
145 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 10,080F1 = (1 + 2x2)
F2 = (1 + 2x)4
F4 = (1 + 2x)8
24
42
12211
422
2164104284276444844484256464
146 0 0 0 2 3 0 0 0 0 0 0 0 0 0 0 0 3360
R
F1 = 1
F2 = (1 + 2x)2
F4 = (1 + 2x)8
22
43
2242741124280444844484256464
147 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 102,520F1 = (1 + 2x)2(1 + 2x6)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)2(1 + 2x2)3
F6 = (1 + 2x)8
14
62
1432
617
2238
670
28312
6178
28316
6288
122
3186289
1824
386164
1824
38636
148 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 102,520
R
F1 = (1 + 2x)2
F2 = (1 + 2x)2
F3 = (1 + 2x)2
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)2
F12 = (1 + 2x)8
14
12
141294 123744129244121484 12149441284441220
149 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 205,040
R
F1 = (1 + 2x)(1 + 2x6)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x)(1 + 2x2)3
F6 = (1 + 2x)8
12
2
62
2232
617
2234
672
2834
6182
2838
6292
122
326297
1426
346166
28640
150 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 205,040F1 = (1 + 2x)
F2 = (1 + 2x)2
F3 = (1 + 2x)
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)2
F12 = (1 + 2x)8
12
2
12
221294
1237
44129244121484
12149
441284441220
151 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 102,520F1 = (1 + 2x6)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x2)3
F6 = (1 + 2x)8
22
62
2232
617
226742834
6182
286296122
326297
28616828640
152 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 102,520
R
F1 = 1
F2 = (1 + 2x)2
F3 = 1
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)2
F12 = (1 + 2x)8
22
12
221294
1237
44129244121484
12149
441284441220
153 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 215,040
R
F1 = (1 + 2x2)(1 + 2x6)
F2 = (1 + 2x)2(1 + 2x3)2
F3 = (1 + 2x2)4
F6 = (1 + 2x)8
22
62
122
32617
226742838
6180
286296122
3106293
2861681426
34638
154 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 215,040F1 = (1 + 2x2)
F2 = (1 + 2x)2
F3 = (1 + 2x2)
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)2
F12 = (1 + 2x)8
22
12
122
129
4 123744129244121484 12149441284441220
155 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 215,040F1 = (1 + 2x6)
F2 = (1 + 2x3)2
F3 = (1 + 2x2)3
F4 = (1 + 2x)2(1 + 2x3)2
F6 = (1 + 2x)6
F12 = (1 + 2x)8
4
62
324
69124
22626
1224
3444
6381272
44632
12132
12232
6912144
441284441220
156 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 215,040
R
F1 = 1
F2 = 1
F3 = 1
F4 = (1 + 2x)2(1 + 2x3)2
F6 = 1
F12 = (1 + 2x)8
4
12
4 1294 123744129244121484 12149441284441220
157 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 80,640
R
F1 = (1 + 2x4)2
F2 = (1 + 2x2)4
F4 = (1 + 2x)8
4424426411214210
4274
4448216444042561426
460
158 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 161,280F1 = (1 + 2x4)
F2 = (1 + 2x2)2
F4 = (1 + 2x)4
F8 = (1 + 2x)8
42
8
2245
811
48852122
438138
822482248128832
159 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 80,640
R
F1 = 1
F2 = 1
F4 = 1
F8 = (1 + 2x)8
828148568140822482248128832
160 3 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 21,504
R
F1 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)8
16
52
11252018588522412535811253561245200116548
161 2 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 64,512F1 = (1 + 2x)2(1 + 2x5)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)7
F10 = (1 + 2x)8
14
2
52
1424
516102
24556
1016
5112
1056
125134
10112
1822
58810134
1828
5241088
281024
162 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 21,504F1 = (1 + 2x)3
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)3
F10 = (1 + 2x)8
16
10
1121010181044101122
10179
2610178212 10100281024
163 2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 64,512
R
F1 = (1 + 2x)2
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)2
F10 = (1 + 2x)8
14
2
10
1424
1010
241044101122
10179
2610178212 10100281024
164 1 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 64,512
R
F1 = (1 + 2x)(1 + 2x5)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)6
F10 = (1 + 2x)8
12
22
52
26512
104
24532
1028
548108812538
10160
1424
51210172
21210100281024
165 1 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 64,512F1 = (1 + 2x)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)
F10 = (1 + 2x)8
12
22
10
261010241044101122
10179
2610178212 10100281024
166 0 0 0 3 0 0 0 1 0 0 0 0 0 0 0 0 21,504F1 = (1 + 2x5)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)5
F10 = (1 + 2x)8
23
52
2658
106
24516
1036
516101041256
10176
261017821210100281024
167 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 64,512
R
F1 = 1
F2 = (1 + 2x)3(1 + 2x5)
F5 = 1
F10 = (1 + 2x)8
23
10
261010241044101122
10179
261017821210100281024
168 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 129,024F1 = (1 + 2x)(1 + 2x2)(1 + 2x5)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)6(1 + 2x2)
F10 = (1 + 2x)8
12
22
52
1225
512104
1422
5361026
5721076125102
10128
1424
5108
10124
14210
5761062
1824
5241012
169 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 129,024
R
F1 = (1 + 2x)(1 + 2x5)
F2 = (1 + 2x)(1 + 2x5)
F4 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)6
F10 = (1 + 2x)6
F20 = (1 + 2x)8
12
4
52
43512
202
42532
2014
548204412538
2080
1442
512
2086
462050442012
170 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 129,024
R
F1 = (1 + 2x)(1 + 2x2)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)(1 + 2x2)
F10 = (1 + 2x)8
12
22
10
1225
1010
1422
1044
101122
10179
26
10178
212
10100
281024
171 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 129,024F1 = (1 + 2x)
F2 = (1 + 2x)(1 + 2x5)
F4 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)
F10 = (1 + 2x)6
F20 = (1 + 2x)8
12
4
10
43106
202
421016
2014
102420442
10192080
2242 1062086462050442012
172 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 129,024
R
F1 = (1 + 2x2)(1 + 2x5)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x2)(1 + 2x)5
F10 = (1 + 2x)8
23
52
1225
58106
24520
1034
532109612538
10160
26532 1016214210
5121094
281024
173 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 129,024F1 = (1 + 2x5)
F2 = (1 + 2x)(1 + 2x5)
F4 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x)5
F10 = (1 + 2x)6
F20 = (1 + 2x)8
2
4
52
4358
102202
42516
1082014
5161016
2044
1256
10162080
2242
1062086
462050442012
174 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 129,024F1 = (1 + 2x2)
F2 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x2)
F10 = (1 + 2x)8
23
10
1225
1010
241044101122 101792610178212 10100281024
175 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 129,024
R
F1 = 1
F2 = (1 + 2x)(1 + 2x5)
F4 = (1 + 2x)3(1 + 2x5)
F5 = 1
F10 = (1 + 2x)6
F20 = (1 + 2x)8
2
4
10
43106
202
421016
2014
102420442
10192080
2242
1062086
462050442012
176 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 368,640
R
F1 = (1 + 2x)(1 + 2x7)
F7 = (1 + 2x)8
12
72
71676471607256725612714614736
177 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 368,640F1 = (1 + 2x)
F2 = (1 + 2x)(1 + 2x7)
F7 = (1 + 2x)
F14 = (1 + 2x)8
12
14
1481432148014128141282
1473
221418
178 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 368,640
R
F1 = (1 + 2x7)
F2 = (1 + 2x)(1 + 2x7)
F7 = (1 + 2x)7
F14 = (1 + 2x)8
2
72
712142740141278014407961480764149612718
1464
221418
179 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 368,640F1 = 1
F2 = (1 + 2x)(1 + 2x7)
F7 = 1
F14 = (1 + 2x)8
2
14
1481432148014128141282
1473
221418
180 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 64,5120F1 = (1 + 2x8)
F2 = (1 + 2x4)2
F4 = (1 + 2x2)4
F8 = (1 + 2x)8
82428138562245
8137
82244882208128122
43830
181 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 64,5120
R
F1 = 1
F2 = 1
F4 = 1
F8 = 1
F16 = (1 + 2x)8
1616716281670161121611216641616
182 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 172,032
R
F1 = (1 + 2x3)(1 + 2x5)
F3 = (1 + 2x)3(1 + 2x5)
F5 = (1 + 2x3)(1 + 2x)5
F15 = (1 + 2x)8
32
52
3458
154
1232
5161524
520156812522
15112
34532
15108
38532
1556
1434
5121512
183 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 172,032F1 = (1 + 2x3)
F2 = (1 + 2x3)(1 + 2x5)
F3 = (1 + 2x)3
F5 = (1 + 2x3)
F6 = (1 + 2x)3(1 + 2x5)
F10 = (1 + 2x3)(1 + 2x)5
F15 = (1 + 2x)3
F30 = (1 + 2x)8
32
10
34104
302
1232 1083012101030342
10113056
62 1016305464
10163028
2262 106306
184 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 172,032F1 = (1 + 2x5)
F2 = (1 + 2x3)(1 + 2x5)
F3 = (1 + 2x5)
F5 = (1 + 2x)5
F6 = (1 + 2x)3(1 + 2x5)
F10 = (1 + 2x3)(1 + 2x)5
F15 = (1 + 2x)5
F30 = (1 + 2x)8
52
6
5862
302
2
516
6
3012
516102
3034
1256
108
3056
62
1016
3054
641016
3028
2262
106306
185 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 172,032
R
F1 = 1
F2 = (1 + 2x3)(1 + 2x5)
F3 = 1
F5 = 1
F6 = (1 + 2x)3(1 + 2x5)
F10 = (1 + 2x3)(1 + 2x)5
F15 = 1
F30 = (1 + 2x)8
6
10
62 1043022 6
1083012
101030342 1011
3056
62 10163054641016
3028
2262
106306
Table 2. Generating functions for the one-dimensional and 7-dimensional IRs for coloring hepteracts (q = 1) for 4 colors.
Table 2. Generating functions for the one-dimensional and 7-dimensional IRs for coloring hepteracts (q = 1) for 4 colors.
[λ]N(A1)N(A3)N(A4)N(A5)N(A7)N(A8)N(A9)N(A10)
16 0 0 01 0 0
15 1 0 01 1 1
14 2 0 02 2 1
13 3 0 02 3 2
12 4 0 03 4 2
11 5 0 03 5 3
10 6 0 04 6 3
9 7 0 04 7 4
8 8 0 051 7 4
14 1 1 02 3 3
13 2 1 03 6 5
12 3 1 04 9 7
11 4 1 05 12 9
10 5 1 06 15 11
9 6 1 07 18 13
8 7 1 081 120 15
12 2 2 06 12 8
11 3 2 07 18 13
10 4 2 010 25 16
9 5 2 011 31 21
8 6 2 0141 137 24
7 7 2 0141 239 27
10 3 3 010 28 20
9 4 3 013 39 27
8 5 3 0161 149 34
7 6 3 0181 156 39
8 4 4 0191 155 36
7 5 4 0211 267 46
6 6 4 0241 273 48
6 5 5 0241 278 54
13 1 1 14 9 9
12 2 1 17 18 16
11 3 1 19 27 23
10 4 1 112 37 30
9 5 1 114 46 37
8 6 1 1171 255 44
7 7 1 1183 558 48
11 2 2 112 36 30
10 3 2 117 56 45
9 4 2 122 77 60
8 5 2 1271 297 75
7 6 2 1313 7111 87
9 3 3 124 88 70
8 4 3 1331 2124 96
7 5 3 1393 7152 119
6 6 3 1433 9166 129
7 4 4 1443 71711 132
6 5 4 1513 920221156
5 5 5 1543 921933171
10 2 2 224 75 57
9 3 2 232 117 91
8 4 2 2451 2165 122
7 5 2 2523 92021 154
6 6 2 2596 1522111166
8 3 3 2481 2189 146
7 4 3 2643 92622 201
6 5 3 2756 1831153240
6 4 4 2876 1835495266
5 5 4 29161213861410297
7 3 3 3713 93032 237
6 4 3 3956 21411116319
5 5 3 3102101304501714354
5 4 4 3117102345162820402
4 4 4 4138153455944330456
Table 3. Generating functions for the binomial colorings of the vertices (q = 8) for one-dimensional IRs of the 8-cube.
Table 3. Generating functions for the binomial colorings of the vertices (q = 8) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11010
28040
3320320
437313131
546471194647119
691028139088972213852
720740597948552074059794855
851107344311770615107315831174193
912459300659651915161245930065965191516
1028900653074253089425042889984971225308861906
11625715497344583809974962625715497344583809974962
1212562875567065121136978106121256285932721012113696094241
13233750783834504229300148849871233750783834504229300148849871
144038807303045625399780438645991240388070211952713997804356178806
1565003434860142353646504358861160586500343486014235364650435886116058
16977872935273906860975020351847385105977872931016186973975020351387480625
1713795944871933252078137742222021879646571379594487193325207813774222202187964657
18183113271146620771933182956842576531615461183113271089874321619182956842570390661781
192293288191579045041618229221962805203235953422932881915790450416182292219628052032359534
2027172601104679810308230271656575432520570557412717260110400461925535727165657543178941093106
21305350757901312578538505305307729144669187569341305350757901312578538505305307729144669187569341
223261598821371763396803511326134394792344579165688332615988213645207951132333261343947922661301616505
2333182648967085223933532504331812029111527294711335103318264896708522393353250433181202911152729471133510
24322145133157474420333016643322137259656650456105184415322145133157403806248865438322137259656642806740494424
252989490904329310130221003214298944969154289160275176131629894909043293101302210032142989449691542891602751761316
2626560397643022444157021925451265601899241818692444349331092656039764302181406660805539326560189924181800986480480083
27226254859185460885384656777972226253849597184975434171952862226254859185460885384656777972226253849597184975434171952862
281850439767413213381676300410605185043502898936770110966723985418504397674132082059483371389381850435028989367140412294430685
Table 4. Generating functions for the binomial colorings of edges (q = 7) for one-dimensional IRs of the 8-cube.
Table 4. Generating functions for the binomial colorings of edges (q = 7) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11000
214040
321601530
4131431285120711274
513689445798411357322579593
6179718686129821262179515891129822963
723649956067208974675552364743597720897539088
82893910524721276027687889028938713873452760280157691
9321959734753775316179169443213321959246156199316179239889735
1032497054656052201322715721990625643249704782855626232271573699877343
112989231824380170346298122165658579585229892317405470058332981221682328226491
12252133297490831881715251871989699732625444252133296410532691241251871990120247195309
1319621374682741244060254196134921233484989379561962137466988364185155119613492129491101473396
141416769496997564229002044141654830826666209884437214167694968431040252489681416548308351788104311581
1595391267226534335414344124953854641196308570351224819539126722477831458020283595385464120726678222923561
166015500298569544153993662343601535731235244289102758981660155002985498220201385551746015357312365881101185952048
17356681204098702555735994657211356677882377176721030402271213356681204098490475804398300875356677882377332607124361022833
1819954275394810936617446090999298199542023864505067799382480016571995427539480870671691960653338519954202386452238537914701133372
Table 5. Generating functions for the binomial colorings of faces (q = 6) for one-dimensional IRs of the 8-cube.
Table 5. Generating functions for the binomial colorings of faces (q = 6) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11000
217030
3502832516
471333243546361225469
517405405126816061703630212854742
64637637972421288829946151751644228812207
71144154282643111101309350811428332131651112152780844
8252694411554530250418661767733252621360494651250486917059594
949933006129016491497935808724807514992931616143925549797157352752701
108894464990395578492888675212130748454888942946674590341868886919926687320706
111440472783586317238393144008373348178394590114404655749637153549941440090890477948652349
12213770312441853780512224213752271869558123835303213770031051943657246139213752552262354430991274
1329269240908200356802347666292684665775569623052621112926923071902421054853647729268476748627548133819831
143719250527778112615739134335371921958419437821456700827337192501836751865245284044993719219927985383328745758341
15440853907156944454398222081566440852750059160355141914968142440853896266243008648327879143440852760944756455945032626186
1648962293141051882905412809207021489622524751529773318383478307394896229281662930067034713894544548962252799495751305458642354391
Table 6. Generating functions for the binomial colorings of cubic cells (q = 5) for one-dimensional IRs of the 8-cube.
Table 6. Generating functions for the binomial colorings of cubic cells (q = 5) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11000
217020
35201219731
473190239555152133050
517658730125191731559369014108134
64670541393418408283044600022354377662001
71147944547821110737977262711282279900171126561130037
8253070289097552250047877926328251418412879773251684384321815
949965338520426373497613947989323644984140421797445649884926827502047
108896911981316427886888430890413955660988885175418967478218892693620617688255
111440638022063010631910143991858406075607173114401202525614681921261440436135220492411513
12213780388559484507191771213742197694885298807844213751077431815470552985213771504232215034965055
1329269801356205901555855379292679061691330856415830322926826801408309293128833729269439420444435412633656
143719279200778578000458180353371919091195188745683667203837192046400308121017459727363719265471000672398359114098
15440855265990629080980664890644440851391239200521016160622915440851878646143680714866519271440854778553508389820473453710
1648962353150087477418358489750594489621924663534229564214473650434896220873318335924169647398045148962336882746820615764869368554
Table 7. Generating functions for the binomial colorings of tesseracts (q = 4) for one-dimensional IRs of the 8-cube.
Table 7. Generating functions for the binomial colorings of tesseracts (q = 4) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11000
214010
327104410
418436175565635422
5219336787855213105791462703
6316843509215441296244997541277130468
745385903112382646563793991546772143430408845
86032761198299557797140701356566302419815945904368147
9730228248478031703689117001293706979787447992726738931053597
1080357795001409842789358018326258727905900475204142480230215709062022
118072061169923281815800180511719478962380059902929049550048067787788823615438
12744350299080496007833741136858948772220225741267190289028028069744218331077926971583
1363372848789530529909508632362542502721236681156324000532533060856199263369069612560717297062
145008288556152866356535055500287217863651434821103050029726175413732759121175008187668616345917852635
15369178651343422851704804726368977568670922245142254944368980084480516795405801544369176128846755941869855695
1625492963441401967651387305634254859500095521515533946430362548600923597874909097205327025492904121555310159685722022
Table 8. Generating functions for the binomial colorings of penteracts (q = 3) for one-dimensional IRs of the 8-cube.
Table 8. Generating functions for the binomial colorings of penteracts (q = 3) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11000
29000
383010
41752311763
5536312959957113129
622066784783907656451186490
7100831126432803455275789776988844
84655630517286980311331260930344145344517
9206159944002155185722872161095026870196198260213
108552744534853721428537045273338256549858374506027325
11329462356516427297042684545365299205065844492326523850764241
1211762768558587306110358322624974281107130601073939311717875989405083
13389788558944470907374644306545546766375177718301384463389149723370338717
1412022008920943629364117278213967002432531173523564889351905112013501431327418719
15346243083080541945354340895758256810378725340991677499412431387346136617589280701403
169343358594200843070436925210976622803392827192532713986707257460989342101964148242810759
17237003980460152929758378235537560933998269545497235550794325997486640218236989945316393042010695
185668492308224642598946637564623338362051324715998056463757918091265854933805668343540316085225671505
Table 9. Generating functions for the binomial colorings of hexeracts (q = 2) for one-dimensional IRs of the 8-cube.
Table 9. Generating functions for the binomial colorings of hexeracts (q = 2) for one-dimensional IRs of the 8-cube.
n1N(A1)N(A2)N(A3)N(A4)
01000
11000
25000
319000
497010
5523070
6336412113100
72349546818412900
8177163115662663449872
91381168202782340001641523
1010815219276480538171276806701
1182876768310641513803602562678809
12610666743298841335339544103515757045
134277230169252630524727391781163859880393
1428291585986191125204352012271654126567032313
15176133023346131124054411135519699197169412236898
1610308004606028238904067488415779340791006017212919
175671131339771477719396400748434804331255584418861048
1829353498288379257149732802732594771938792329064956439595
19143105988701069129121616153798129891161595220142191025038525
20658063307862705607198937700202609605218691334655293632298405
212858457998939006268301573379068126901576394719482850441762739161
2211746243530321863111712736350720191119145750748842111724027454457595
2345730471397391867439368446921012954399130680166176245671434719950791
24168913836578625999163578906149612869163719543899932333168763207445340396
25592737125111757011577583393796842496577931661515814991592367656948196051
261978559832758151399193739346265546250219382219873217028461977687567195144310
276289941346098637974618284064001027251861847371934736898646287956989620704957
2819065234710132878894187980430923955619331880222623753317053219060880013556511457
2955155926406439783261545159704792441528625452487203773125404455146698500000142941
30152448370819991091907150975194909849530476150993490533551033564152429470569138848128
31402931631210964655490399668880173423084086399705237549199768826402894182587955370920
321019267441120735541170101230831623329871833510123782371931546642521019195601065459214731
Table 10. Generating function for the second IR with dimension 672 for (D2O)8 where three bosnic colors are shown as ordered partitions of [16] into 3 parts for 3 colors.
Table 10. Generating function for the second IR with dimension 672 for (D2O)8 where three bosnic colors are shown as ordered partitions of [16] into 3 parts for 3 colors.
N(Γ672-2)[16]N(Γ672-2)[16]
110 5 1198 2 6
29 6 1627 3 6
38 7 11166 4 6
37 8 11425 5 6
26 9 11164 6 6
15 10 1623 7 6
111 3 2192 8 6
510 4 221 9 6
129 5 238 1 7
198 6 2227 2 7
227 7 2626 3 7
196 8 2995 4 7
125 9 2994 5 7
54 10 2623 6 7
13 11 2222 7 7
111 2 331 8 7
810 3 337 1 8
249 4 3196 2 8
468 5 3465 3 8
627 6 3604 4 8
626 7 3463 5 8
465 8 3192 6 8
244 9 331 7 8
83 10 326 1 9
12 11 3125 2 9
510 2 4244 3 9
249 3 4243 4 9
608 4 4122 5 9
997 5 421 6 9
1166 6 415 1 10
995 7 454 2 10
604 8 483 3 10
243 9 452 4 10
52 10 411 5 10
110 1 513 2 11
129 2 512 3 11
468 3 5
997 4 5
1426 5 5
1425 6 5
994 7 5
463 8 5
122 9 5
11 10 5
29 1 6
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Balasubramanian, K. Recursive Symmetries: Chemically Induced Combinatorics of Colorings of Hyperplanes of an 8-Cube for All Irreducible Representations. Symmetry 2023, 15, 1031. https://doi.org/10.3390/sym15051031

AMA Style

Balasubramanian K. Recursive Symmetries: Chemically Induced Combinatorics of Colorings of Hyperplanes of an 8-Cube for All Irreducible Representations. Symmetry. 2023; 15(5):1031. https://doi.org/10.3390/sym15051031

Chicago/Turabian Style

Balasubramanian, Krishnan. 2023. "Recursive Symmetries: Chemically Induced Combinatorics of Colorings of Hyperplanes of an 8-Cube for All Irreducible Representations" Symmetry 15, no. 5: 1031. https://doi.org/10.3390/sym15051031

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