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

Krishnan Balasubramanian

doi:10.33187/jmsm.471940

EN

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 M\"{o}bius 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.

Keywords

Combinatorial Enumerations,Colorings of hypercubes,Character Tables of hypercubes,Pólya Theory for all charters,Character cycle indices for all hyperplanes

References

[1] R. Carbo-Dorca, Boolean hypercubes and the structure of vector spaces, J. Math. Sciences and Model. 1(1) (2018), 1-14.
[2] R. Carbo-Dorca, N-dimensional Boolean hypercubes and the goldbach conjecture, J. Math. Chem. 54(6) (2016), 1213-1220. https://doi.org/10.1007/s10910-016-0628-5
[3] R. Carbo-Dorca, DNA, unnatural base pairs and hypercubes, J. Math. Chem. 56(5) (2018), 1353-1356. https://doi.org/10.1007/s10910-018-0866-9
[4] R. Carbo-Dorca, About Erdös discrepancy conjecture, J. Math. Chem. 54(3) (2016), 657-660. https://doi.org/10.1007/s10910-015-0585-4
[5] R. Carbo-Dorca, Boolean hypercubes as time representation holders, J. Math. Chem. 56(5) (2018), 1349-1352. https://doi.org/10.1007/s10910-018- 0865-x
[6] A. A. Gowen, C. P. O’Donnell, P. J. Cullen, S. E. J. Bell, Recent applications of chemical imaging to pharmaceutical process monitoring and quality control, European Journal of Pharmaceutics and Biopharmaceutics 69(1) (2008), 10-22.
[7] P. G. Mezey, Similarity analysis in two and three dimensions using lattice animals and ploycubes, J. Math. Chem. 11(1) (1992), 27-45.
[8] A. Frolov, E. Jako, P. G. Mezey, Logical models for molecular shapes and their families, J. Math. Chem. 30(4) (2001), 389-409.
[9] P. G. Mezey, Some dimension problems in molecular databases, J. Math. Chem. 45(1) (2009), 1-6.
[10] P. G. Mezey, Shape similarity measures for molecular bodies: A three-dimensional topological approach in quantitative shape-activity relations, J. Chem. Inf. Comput. Sci. 32(6) (1992), 650-656.

[11] K. Balasubramanian, Combinatorial multinomial generators for colorings of 4D-hypercubes and their applications, J. Math. Chem. 56(9) (2018), 2707-2723.
[12] W. K. Clifford, Mathematical papers, Macmillan and Company, London, 1882.
[13] W. K. Clifford, On the types of compound statement involving four classes, Memoirs of the Literary and Philosophical Society of Manchester 16 (1877), 88-101.
[14] G. Polya, R. C. Read, Combinatorial enumeration of groups, graphs and chemical compounds, Springer, New York, 1987.
[15] G. Polya, Kombinatorische anzahlbestimmugen f¨ur gruppen, graphen und chemische verbindugen, Acta. Math. 68(1) (1937), 145-254.
[16] J. H. Redfield, The theory of group-reduced distributions, American Journal of Mathematics 49(3) (1927), 433-455.
[17] G. P´olya, Sur les types des propositions compos´ees, The Journal of Symbolic Logic 5(3) (1940), 98-103.
[18] M. A. Harrison, R. G. High, On the cycle index of a product of permutation groups, Journal of Combinatorial Theory 4(3) (1968), 277-299.
[19] D. C. Banks, S. A. Linton, P. K. Stockmeyer, Counting cases in substitope algorithms, IEEE Transactions on Visualization and Computer Graphics 10(4) (2004), 371-384.
[20] D. C. Banks, P. K. Stockmeyer, DeBruijn counting for visualization algorithms, T. M´oller, B. Hamann, R. D. Russell (editors), Mathematical foundations of scientific visualization, computer graphics and massive data exploration, Springer, Berlin, 2009, pp. 69-88.
[21] W. Y. C. Chen, Induced cycle structures of the hyperoctahedral group, SIAM J. Discrete Math. 6(3) (1993), 353-362.
[22] G. M. Ziegler, Lectures on polytopes (graduate texts in mathematics; 152), Springer-Verlag, 1994.
[23] P. W. H. Lemmens, P´olya theory of hypercubes, Geometriae Dedicata 64(2) (1997), 145-155.
[24] P. Bhaniramka, R. Wenger, R. Crawfis, Isosurfacing in higher dimensions, Proceedings of IEEE Visualization 2000, (2000), 267-273.
[25] O. Aichholzer, Extremal properties of 0/1-polytopes of dimension 5, G. Kalai, G. M. Ziegler (editors), Polytopes - combinatorics and computation, Birkhauser, Basel, 2000, pp. 111-130.
[26] R. Perez-Aguila, Enumerating the configurations in the n-dimensional polytopes through P´olya’s counting and a concise representation, 3rd International Conference on Electrical and Electronics Engineering, (2006), 1-4.
[27] M. Liu, K. E. Bassler, Finite size effects and symmetry breaking in the evolution of networks of competing Boolean nodes, Journal of Physics A: Mathematical and Theoretical 44(4) (2010), 045101.
[28] R. Perez-Aguila, Towards a new approach for volume datasets based on orthogonal polytopes in four-dimensional color space, Engineering Letters 18(4) (2010), 326-340.
[29] W. Y. C. Chen, P. L. Guo, Equivalence classes of full-dimensional 0/1-polytopes with many vertices, (2011), arXiv:1101.0410v1 [math.CO].
[30] N. G. de Bruijn, Enumeration of tree-shaped molecules, W. T. Tutte (editor), Recent progress in combinatorics: proceedings of the 3rd Waterloo conference on combinatorics, Academic Press, New York, 1969, pp. 59-68.
[31] F. Harary, E. M. Palmer, Graphical enumeration, Academic Press, New York, 1973.
[32] I. G. Macdonald, E. M. Palmer, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.
[33] A. T. Balaban, Enumeration of isomers, D.Bonchev, D. H. Rouvray (editors), Chemical graph theory: introduction and fundamentals, Abacus Press/Gordon and Breach Science Publishers, New York, 1991, pp. 177-234.
[34] C. J. O. Reichhardt, K. E. Bassler, Canalization and symmetry in Boolean models for genetic regulatory networks, Journal of Physics A: Mathematical and Theoretical 40(16) (2007), 4339.
[35] K. Balasubramanian, Combinatorial enumeration of ragas (scales of integer sequences) of Indian music, Journal of Integer Sequences 5 (2002), Article 02.2.6.
[36] K. Balasubramanian, Applications of combinatorics and graph theory to spectroscopy and quantum chemistry, Chem. Rev. 85(6) (1985), 599-618.
[37] K. Balasubramanian, The symmetry groups of nonrigid molecules as generalized wreath-products and their representations, J. Chem. Phys. 72(1) (1980), 665-677.
[38] K. Balasubramanian, Relativistic double group spinor representations of nonrigid molecules, J. Chem. Phys. 120(12) (2004), 5524-5535.
[39] K. Balasubramanian, Generalization of de Bruijn’s extension of P´olya’s theorem to all characters, J. Math. Chem. 14(1) (1993), 113-120.
[40] K. Balasubramanian, Generalization of the Harary-Palmer power group theorem to all irreducible representations of object and color groups-color combinatorial group theory, J. Math. Chem. 52(2) (2014), 703-728.
[41] R. Wallace, Spontaneous symmetry breaking in a non-rigid molecule approach to intrinsically disordered proteins, Molecular BioSystems 8(1) (2012), 374-377.
[42] R. Wallace, Tools for the future: hidden symmetries, Computational Psychiatry, Springer, Cham, 2017, pp. 153-165.
[43] M. R. Darafsheh, Y. Farjami, A. R. Ashrafi, Computing the full non-rigid group of tetranitrocubane and octanitrocubane using wreath product, MATCH Commun. Math. Comput. Chem. 54(1) (2005),53-74.
[44] R. Foote, G. Mirchandani, D. Rockmore, Two-dimensional wreath product group-based image processing, Journal of Symbolic Computation 37(2) (2004), 187-207.
[45] K. Balasubramanian, A generalized wreath product method for the enumeration of stereo and position isomers of polysubstituted organic compounds, Theoret. Chim. Acta 51(1) (1979), 37-54.
[46] K. Balasubramanian, Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy, International Journal of Quantum Chemistry 20(6) (1981), 1255-1271.
[47] K. Balasubramanian, Enumeration of the isomers of the gallium arsenide clusters (GamAsn), Chemical Physics Letters 150(1-2) (1988), 71-77.
[48] K. Balasubramanian, Nuclear-spin statistics of C60, C60H60 and C60D60, Chemical Physics Letters 183(3-4) (1991), 292-296.
[49] K. Balasubramanian, Group theoretical analysis of vibrational modes and rovibronic levels of extended aromatic C48N12 azafullerene, Chemical Physics Letters 391(1-3) (2004), 64-68.
[50] K. Balasubramanian, Group theory and nuclear spin statistics of weakly-bound (H2O)n; (NH3)n; (CH4)n; and NH+ 4(NH3)n, J. Chem. Phys. 95(11) (1991), 8273-8286.
[51] K. Balasubramanian, Generators of the character tables of generalized wreath product groups, Theoretica Chimica Acta 78(1) (1990), 31-43.
[52] X. Liu, K. Balasubramanian, Computer generation of character tables of generalized wreath product groups, Journal of Computational Chemistry 11(5) (1990), 589-602.
[53] K. Balasubramanian, Multinomial combinatorial group representations of the octahedral and cubic symmetries, Journal of Mathematical Chemistry 35(4) (2004), 345-365.
[54] K. Balasubramanian, Enumeration of internal rotation reactions and their reaction graphs, Theoretica Chimica Acta 53(2) (1979), 129-146.
[55] K. Balasubramanian, A method for nuclear-spin statistics in molecular spectroscopy, J. Chem. Phys. 74(12) (1981), 6824-6829.
[56] K. Balasubramanian, Operator and algebraic methods for NMR spectroscopy. II. NMR projection operators and spin functions, J. Chem. Phys. 78(11) (1983), 6369-6376.
[57] K. Balasubramanian, M. Randic, The characteristic polynomials of structures with pending bonds, Theoretica Chimica Acta 61(4) (1982), 307-323.
[58] S. C. Basak, D. Mills, M. M. Mumtaz, K. Balasubramanian, Use of topological indices in predicting aryl hydrocarbon receptor binding potency of dibenzofurans: A hierarchical QSAR approach, Indian Journal of Chemistry-Section A 42A (2003), 1385-1391.
[59] T. Ruen, Free Public Domain Work, available to anyone to use for any purpose at https://commons.wikimedia.org/wiki/File:5-cube_t024.svg
[60] N. G. de Bruijn, Color Patterns that are invariant under permutation of colors, Journal of Combinatorial Theory 2(4) (1967), 418-421.
[61] K. Balasubramanian, Computational multinomial combinatorics for colorings of 5D-hypercubes for all irreducible representations and applications, J. Math. Chem. (2018), https://doi.org/10.1007/s10910-018-0978-2
[62] J. M. Price, M. W. Crofton, Y. T. Lee, Vibrational spectroscopy of the ammoniated ammonium ions NH4 +(NH3)n(n = 1􀀀10), Journal of Physical Chemistry 95(6) (1991), 2182-2195.
[63] K. Balasubramanian, Enumeration of stable stereo and position isomers of polysubstitued alcohols, ANNALS of the New York Academy of Sciences 319(1) (1979), 33-36.
[64] K. Balasubramanian, Nonrigid group theory, tunneling splittings, and nuclear spin statistics of water pentamer: (H2O)5, The Journal of Physical Chemistry A 108(26) (2004), 5527-5536.
[65] H. S. M. Coxeter, Regular polytopes, Dover Publications, New York, 1973.
[66] J. W. Kennedy, M. Gordon, Graph contraction and a generalized M¨obius inversion, Annals of the New York Academy of Sciences 319(1) (1979), 331-348.
[67] V. Krishnamurthy, Combinatorics: theory and applications, Ellis Harwood, New York, 1986.
[68] K. Balasubramanian, Generating functions for the nuclear spin statistics of nonrigid molecules, J. Chem. Phys. 75(9) (1981), 4572-4585.
[69] K. Balasubramanian, Operator and algebraic methods for NMR spectroscopy. I. Generation of NMR spin species, J. Chem. Phys. 78(11) (1983), 6358-6368.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Krishnan Balasubramanian ^*
United States

Publication Date

December 30, 2018

Submission Date

October 18, 2018

Acceptance Date

November 29, 2018

Published in Issue

Year 2018 Volume: 1 Number: 3

DOI

https://doi.org/10.33187/jmsm.471940

IZ

https://izlik.org/JA35CH59NA

APA

Balasubramanian, K. (2018). Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications. Journal of Mathematical Sciences and Modelling, 1(3), 158-180. https://doi.org/10.33187/jmsm.471940

AMA

1.Balasubramanian K. Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications. Journal of Mathematical Sciences and Modelling. 2018;1(3):158-180. doi:10.33187/jmsm.471940

Chicago

Balasubramanian, Krishnan. 2018. “Computational Enumeration of Colorings of Hyperplanes of Hypercubes for All Irreducible Representations and Applications”. Journal of Mathematical Sciences and Modelling 1 (3): 158-80. https://doi.org/10.33187/jmsm.471940.

EndNote

Balasubramanian K (December 1, 2018) Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications. Journal of Mathematical Sciences and Modelling 1 3 158–180.

IEEE

[1]K. Balasubramanian, “Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 3, pp. 158–180, Dec. 2018, doi: 10.33187/jmsm.471940.

ISNAD

Balasubramanian, Krishnan. “Computational Enumeration of Colorings of Hyperplanes of Hypercubes for All Irreducible Representations and Applications”. Journal of Mathematical Sciences and Modelling 1/3 (December 1, 2018): 158-180. https://doi.org/10.33187/jmsm.471940.

JAMA

1.Balasubramanian K. Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications. Journal of Mathematical Sciences and Modelling. 2018;1:158–180.

MLA

Balasubramanian, Krishnan. “Computational Enumeration of Colorings of Hyperplanes of Hypercubes for All Irreducible Representations and Applications”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 3, Dec. 2018, pp. 158-80, doi:10.33187/jmsm.471940.

Vancouver

1.Krishnan Balasubramanian. Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications. Journal of Mathematical Sciences and Modelling. 2018 Dec. 1;1(3):158-80. doi:10.33187/jmsm.471940