Enumeration of Stereoisomers of Chiral and Achiral Derivatives of Monocyclic Cycloalkanes Having Heteromorphous Alkyl Substituents with Distinct Length k

DOI: 10.4236/cc.2019.73006, PP. 72-93

Keywords: Enumeration, Stereoisomer, Chirality, Achirality, Monocyclic Cycloalkane, Order of Alkyl Trees, Polyalkylation

Abstract:

A combinatorial method based on the determination of the averaged weight of permutations controlling the chirality/achirality fittingness of 2n substitution sites of the monocyclic cycloalkane allows to obtain generalized functional equations for direct enumeration of enantiomers pairs and achiral skeletons of any derivatives of monocyclic cycloalkanes having heteromorphic alkyl substituents with the distinct length k with the empirical formula , wherein at least two alkyl groups？？of the distinct size ？each. ？is the number of alkyl radicals ？of the system？？verifying the relation

References

[1]	Hansen, P.J. and Jurs, P.C. (1988) Chemical Applications of Graph Theory. Isomer Enumeration. Journal of Chemical Education, 65, 662-664. https://doi.org/10.1021/ed065p661
[2]	Henze, H.R. and Blair, C.M.J. (1931) Chem. Soc., 53, (a) 3042-3046, (b) 3077-3085.
[3]	Pólya, G. (1937) Kombinatorische Abzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica, 68, 145-254.
[4]	Burnside, W. (1911) Theory of Groups of Finite Order. 2nd Edition, Cambridge Univ. Press, London.
[5]	de Bruijn, N.J. (1964) Polya’s Theory of Counting. In: Beckenbach, E.F., Ed., Applied Combinatorial Mathematics, John Wiley, New York, Chapter 5.
[6]	Cyvin, S.J., Brunvoll, J.B., Cyvin, N. and Brendsdal, E. (1996) Enumeration of Isomers and Conformers: A Complete Mathematical Solution for Conjugated Polyene Hydrocarbons. Advances in Molecular Structure Research, 2, 213-245. https://doi.org/10.1016/S1087-3295(96)80021-1
[7]	Balasubramanian, K. (1979) A Generalized Wreath Product Method for the Enumeration of Stereo and Position Isomers of Poly-Substituted Organic Compounds. Theoretica Chimica Acta, 51, 37-54. https://doi.org/10.1007/BF02399129
[8]	Balaban, A.T. (1978) Chemical Graphs. XXXII. Constitutional and Steric Isomers of Substituted Cycloalkanes. Croatica Chemica Acta, 51, 35-42. Balaban, A.T., Kennedy, J.W. and Quintas, L.V. (1988) The Number of Alkanes Having N Carbon and a Longest Chain of Length D, an Application of a Theorem of Polya. Journal of Chemical Education, 65, 304-313. https://doi.org/10.1021/ed065p304
[9]	Fowler, P.W. (1995) Isomer Counting Using Point Group Symmetry. Journal of the Chemical Society, Faraday Transactions, 91, 2241-2247. https://doi.org/10.1039/ft9959102241
[10]	Hässelbarth, W. and Ruch, E. (1973) Classification of Rearrangement Mechanisms by Means of Double Cosets and Counting Formulas for Numbers of Classes. Theoretica Chimica Acta, 29, 259-268. https://doi.org/10.1007/BF00529052
[11]	Harary, F. and Palmer, E. (1973) Graphical Enumeration. Academic Press, New York. https://doi.org/10.1016/B978-0-12-324245-7.50005-8
[12]	Kerber, A. (1975) On Graphs and Their Enumeration, Part 1. Math. Chem. Comm., 1, 5-10; (1976) Part II, 2, 17-34. Kerber, A. (1979) Counting Isomers. In: Hinze, J., Ed., The Permutation Group in Physics and Chemistry, Springer-Verlag, Berlin, 1-18.
[13]	Bytautas, L. and Klein, D.J. (1999) Alkane Isomer Combinatorics: Stereostructure Enumeration and Graph Invariant and Molecular Property Distributions. The Journal for Chemical Information and Computer Scientists, 39, 803-808. https://doi.org/10.1021/ci990021g
[14]	Lloyd, E.K. (1992) Marks of Permutation Groups and Isomers Enumeration. Journal of Mathematical Chemistry, 11, 207-222. https://doi.org/10.1007/BF01164205
[15]	Fujita, S. (2000) Systematic Enumeration of Nonrigid Isomers with Given Ligand Symmetries. Journal of Chemical Information and Computer Sciences, 40, 135-146. https://doi.org/10.1021/ci990072d
[16]	Nemba, R.M. and Fah, M. (1997) On the Application of Sieve Formula to the Enumeration of the Stable Stereo and Position Isomers of Deoxycyclitols. Journal of Chemical Information and Computer Sciences, 4, 722-725. https://doi.org/10.1021/ci960129l
[17]	Nemba, R.M. and Ngouhouo, F. (1994) On the Enumeration of Chiral and Achiral Skeletons of Position Isomers of Homosubstituted Monocyclic Cycloalkanes with a Ring Size N (Odd or Even). Tetrahedron, 50, 6663-6670. https://doi.org/10.1016/S0040-4020(01)89694-0
[18]	Nemba, R.M. and Ngouhouo, F. (1994) New Journal of Chemistry, 18, 1175-1182.
[19]	Nemba, R.M. and Balaban, A.T. (1998) Algorithm for the Direct Enumeration of Chiral and Achiral Skeletons of Homosubstituted Derivatives of a Monocyclic Cycloalkane with a Large Ring Size N. Journal of Chemical Information and Computer Sciences, 38, 1145-1150. https://doi.org/10.1021/ci980079f
[20]	Nemba, R.M. (1996) Solution Générale du problème de dénombrement des stéréoisomères d’un cycloalcane homosubstitué. Comptes rendus de l’Académie des Sciences, Série II b, 323, 773-779.
[21]	Nemba, R.M. and Emadak, A. (2002) Direct Enumeration of Chiral and Achiral Graphs of a Polyheterosubstituted Monocyclic Cycloalkane. Journal of Integer Sequences, 5, Article 02.1.6.
[22]	Nemba, R.M. and Emadak, A. (2002) Algorithme de dénombrement des graphes chiraux et achiraux d’un cycloalcane monocyclique hétérosubstitué de formule brute CnHm1Xm2Ym3. Comptes Rendus Chimie, 5, 533-538. https://doi.org/10.1016/S1631-0748(02)01400-5
[23]	Nemba, R.M. and Balaban, A.T. (2002) Enumeration of Chiral and Achiral Isomers of an n-Membered Ring with n Homomorphic Alkyl Groups. Match, 46, 235-250.
[24]	Emadak, A., Nemba, R.M., Patouossa, I., Ndassa, I.M. and Makon, B.T. (2019) Enumeration of Stereoisomers of Chiral and Achiral Derivatives of Monocyclic Cycloalkanes Having Heteromorphous Alkyl Substituents with the Same Length K. International Journal of Chemical Science, 3, 43-53.
[25]	Emadak, A. (2014) Dénombrement des monocycloalcanes hétéropolysubstitués Tome 1: Système ayant une hétéropolysubstitution d’ordre binaire, ternaire, quaternaire, quintuplet et sextuplet. Presse Académique Francophone.
[26]	Robinson, R.W., Harary, F. and Balaban, A.T. (1976) The Number of Chiral and Achiral Alkanes and Monosubstituted Alkanes. Tetrahedron, 32, 355-361. https://doi.org/10.1016/0040-4020(76)80049-X

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