Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications

Abstract

We obtain the generating functions for the combinatorial enumeration of colorings of all hyperplanes of hypercubes for all irreducible representations of the hyperoctahedral groups. The computational group theoretical techniques involve the construction of generalized character cycle indices of all irreducible representations for all hyperplanes of the hypercube using the Möbius function, polynomial generators for all cycle types and for all hyperplanes. This is followed by the construction of the generating functions for colorings of all (n-q)hyperplanes of the hypercube, for example, vertices (q=5), edges (q=4), faces (q=3), cells (q=2) and tesseracts (q=4) for a 5D-hypercube. Tables are constructed for the combinatorial numbers for coloring all hyperplanes of 5D-hypercubes for 36 irreducible representations. Applications to chirality, chemistry and biology are also pointed out.T (g; π) = a ik (1 ≤ i ≤ 2) , (1 ≤ k ≤ 5) (2.1)

Numerical Analysis of the Ground-State Magnetic Moments of ${}^{143,145,147}{\rm{Sm}}$ Isotopes

“…The results obtained reveal that spin interactions allow to successfully explain the spin polarization phenomenon observed in odd-mass nuclei and its effects on ground-state magnetic properties [1][2][3]. The applications in physics have been mostly in the field of general and differential equation [42][43][44].…”

Combinatorics of Supergiant Fullerenes: Enumeration of Polysubstituted Isomers, Chirality, Nuclear Magnetic Resonance, Electron Spin Resonance Patterns, and Vibrational Modes from C70to C150000

“…As DDVS and/or SRWS provides the equivalence classes of partitions of vertices, one can treat members of each shell independent of other shells that result in simplifications owing to the decomposition into partitions of vertices. We invoke the Möbius inversion technique, which has been shown to be quite powerful in the context of hypercube enumerations considered by the author recently. , The principal element of the technique is that, if we know the group action on a smaller set of objects, then the action of the same group on a larger set can be determined through a polynomial Möbius sum involving divisors of the number of elements in the larger set relative to the smaller one. With regard to the permutations of the vertices, if the cycle types for the permutations for the group action on the smaller set is known, the Möbius inversion provides a technique to obtain the cycle types and hence the cycle index polynomial of the larger set.…”

Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C80, C180, and C240

“…Fullerenes are especially interesting candidates for the edge colorings, as they contain a network of alternating arrangements of single and double bonds, which results in a large number of resonance structures or conjugated circuits [43][44][45][46][47][48][49][50][51][52][53], especially for giant on the smaller sets are known, then the technique can be used to obtain the cycle types of a larger set of objects. The application of the technique results in the inversion of a polynomial generating function from one set to the other through the divisors, which we used extensively in the context of hypercube enumerations for coloring the various hyperplanes for all of the irreducible representations of the hyperoctahedral group [55]. That is, the coefficient of x q in the inverted polynomial generating function Q p (x) shown below obtained from a known polynomial F d (x) generates the permutational cycle types for a larger set from a smaller set-for example, the edges of a 7D hypercube from the permutational matrix types for the hexeracts of the 7D hypercube [55].…”

“…The application of the technique results in the inversion of a polynomial generating function from one set to the other through the divisors, which we used extensively in the context of hypercube enumerations for coloring the various hyperplanes for all of the irreducible representations of the hyperoctahedral group [55]. That is, the coefficient of x q in the inverted polynomial generating function Q p (x) shown below obtained from a known polynomial F d (x) generates the permutational cycle types for a larger set from a smaller set-for example, the edges of a 7D hypercube from the permutational matrix types for the hexeracts of the 7D hypercube [55].…”