Abstract
Combined-permutation representations (CPRs) for characterizing \({\varvec{I}}_{h}\)-skeletons (a dodecahedral skeleton 1 having 20 vertices, an icosahedral skeleton 2 having 12 vertices, and a \(\hbox {C}_{{60}}\)-fullerene skeleton 3 having 60 vertices) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to give the CPR of degree 22 (= 20 + 2) for 1, the CPR of degree 14 (= 12 + 2) for 2, and the CPR of degree 62 (= 60 + 2) for 3. Mark tables (tables of marks) of these CPRs are different in the sequence of subgroups from each other when they are produced as primary mark tables by the GAP system. On the other hand, the GAP functions MarkTableforUSCI and constructUSCITable, which have been previously developed to systematize the concordant construction of a standard mark table and a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table, are capable of constructing the standard mark table and the standard USCI-CF table even if we start from any of these CPRs. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (Fujita in Symmetry and combinatorial enumeration in chemistry, Springer, Berlin, 1991). Construction of the CPR of degree 7 (= 5 + 2) for characterizing the \({\varvec{I}}_{h}\) group is also discussed by starting from the alternating group \(\hbox {A}_{{5}}\) isomorphic to the point group \({\varvec{I}}\).
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References
W. Burnside, Theory of Groups of Finite Order, 2nd edn. (Cambridge University Press, Cambridge, 1911)
S. Fujita, Subduction of coset representations. An application to enumeration of chemical structures. Theor. Chim. Acta 76, 247–268 (1989)
S. Fujita, Systematic enumerations of highly symmetric cage-shaped molecules by unit subduced cycle indices. Bull. Chem. Soc. Jpn. 62, 3771–3778 (1989)
S. Fujita, Subduction of coset representations. An application to enumeration of chemical structures with achiral and chiral ligands. J. Math. Chem. 5, 121–156 (1990)
S. Fujita, Systematic enumeration of high symmetry molecules by means of unit subduced cycle indices with and without chirality fittingness. Bull. Chem. Soc. Jpn. 63, 203–215 (1990)
S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry (Springer, Berlin, 1991)
S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers (University of Kragujevac, Faculty of Science, Kragujevac, 2007)
S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds (University of Kragujevac, Faculty of Science, Kragujevac, 2013)
S. Fujita, Chirality fittingness of an orbit governed by a coset representation. Integration of point-group and permutation-group theories to treat local chirality and prochirality. J. Am. Chem. Soc. 112, 3390–3397 (1990)
S. Fujita, Unit subduced cycle indices with and without chirality fittingness for \(\varvec {I}_{h}\) group. An application to systematic enumeration of dodecahedrane derivatives. Bull. Chem. Soc. Jpn. 63, 2759–2769 (1990)
S. Fujita, Computer-oriented representations of point groups and cycle indices with chirality fittingness (CI-CFs) calculated by the gap system. Enumeration of three-dimensional structures of ligancy 4 by Fujita’s proligand method. MATCH Commun. Math. Comput. Chem. 76, 379–400 (2016)
S. Fujita, Computer-oriented representations of \(\mathbf{O}_{{h}}\)-skeletons for supporting combinatorial enumeration by Fujita’s proligand method. GAP calculation of cycle indices with chirality fittingness (CI-CFs). MATCH Commun. Math. Comput. Chem. 77, 409–442 (2017)
S. Fujita, Hierarchical enumeration based on skeletons of ligancy 6 by using combined-permutation representations. Part 1. Cyclopropane derivatives. MATCH Commun. Math. Comput. Chem. 79, 103–142 (2018)
S. Fujita, Soccerane derivatives of given symmetries. Systematic enumeration by means of unit subduced cycle indices. Bull. Chem. Soc. Jpn. 64, 3215–3223 (1991)
S. Fujita, Concordant generation of mark tables and USCI-CF (unit subduced cycle indices with chirality fittingness) tables on the basis of combined-permutation representations. MATCH Commun. Math. Comput. Chem. 82, 295–326 (2019)
S. Fujita, Standardization of mark tables and USCI-CF (unit subduced cycle indices with chirality fittingness) tables derived from different \({\varvec {O}}_{{h}}\)-skeletons. MATCH Commun. Math. Comput. Chem. 82, 327–373 (2019)
S. Fujita, Enumeration of digraphs with a given automorphism group. J. Math. Chem. 12, 173–195 (1993)
S. Fujita, Generalization of partial cycle indices and modified bisected mark tables for combinatorial enumeration. Bull. Chem. Soc. Jpn. 73, 329–339 (2000)
S. Fujita, Enumeration of three-dimensional structures derived from a dodecahedrane skeleton under a restriction condition. I. The restricted-fixed-point-matrix (RFPM) method based on restricted subduced cycle indices. J. Comput. Chem. Jpn. 11, 131–139 (2012)
S. Fujita, Enumeration of three-dimensional structures derived from a dodecahedrane skeleton under a restriction condition. II. The restricted-partial-cycle-index (RPCI) method based on restricted subduced cycle indices. J. Comput. Chem. Jpn. 11, 140–148 (2012)
S. Fujita, Stereogenicity revisited. Proposal of holantimers for comprehending the relationship between stereogenicity and chirality. J. Org. Chem. 69, 3158–3165 (2004)
S. Fujita, Pseudoasymmetry, stereogenicity, and the RS-nomenclature comprehended by the concepts of holantimers and stereoisograms. Tetrahedron 60, 11629–11638 (2004)
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Appendices
Appendix 1: Source Code 1 (SubIh6A.gap)
The CPR I_dod corresponding to the point group \({\varvec{I}}\) without reflections are generated by using a set of generators gen_dodx, which is a part of the set of generators gen_dod for generating the CPR Ih_dod corresponding to the point group \({\varvec{I}}_{h}\) with reflections.
The primary mark table tom_Ih_dod calculated from the CPR Ih_dod is characterized by \(\mathrm {SSG}_{{\varvec{I}}_{h}}^{dod}\) (Eq. 3). In order to convert the mark table into the standard mark table based on \(\mathrm {SSG}_{{\varvec{I}}_{h}}\) (Eq. 1) and to assure the concordance between the two tables, the alignment of the list of subgroups derived from the SSG (\(\mathrm {SSG}_{{\varvec{I}}_{h}}^{dod}\), Eq. 3) is changed to give a list of sets of generators gen[1]–gen[22]. Note that, for example, the row of gen[4] corresponds to \(\overbrace{\underbrace{{\varvec{C}}_{i}}_{4}}^{2}\) in Eq. 3.
Source Code 1 (SubIh6A.gap)
Appendix 2: Source Code 2 (enum-IhdodX.gap)
The following code (Source Code 2: enum-IhdodX.gap) shows the symmetry-itemized enumeration of dodecahedral derivatives on the basis of the PCI-CF method.
Let us select 20 proligands for the dodecahedral skeleton 1 from the ligand inventory \({\varvec{L}}\) shown in Eq. 72, where the uppercase letters A and B represent achiral proligands, while a pair of symbols \(\mathrm {p}/\mathrm {P}\) represents a pair of enantiomeric proligands when detached (e.g., p/\(\overline{\mathrm{p}}\) = p/P = R-CFClBr/S-CFBrCl). The uppercase letter P is used in place of the symbol \(\overline{\mathrm{p}}\) to simplify the source code.
Source Code 2 (enum-IhdodX.gap)
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Fujita, S. Symmtry-itemized enumeration of compounds derived from different \({\varvec{I}}_{h}\)-skeletons by means of combined-permutation representations and newly-developed GAP functions. J Math Chem 58, 1364–1408 (2020). https://doi.org/10.1007/s10910-020-01132-3
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DOI: https://doi.org/10.1007/s10910-020-01132-3