Journal of Mathematical Chemistry

, Volume 42, Issue 2, pp 215–263 | Cite as

Sphericities of Double Cosets, Double Coset Representations, and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

  • Shinsaku Fujita

The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a d , c d , or b d ) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita’s proligand method (S. Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.


coset representation double coset representation enumeration sphericity stereoisomer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pólya G., (1937) Acta Math. 68: 145–254CrossRefGoogle Scholar
  2. 2.
    Pólya G., Read R.C., (1987) Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer-Verlag, New YorkGoogle Scholar
  3. 3.
    Harary F., Palmer E.M., (1973) Graphical Enumeration. Academic Press, New YorkGoogle Scholar
  4. 4.
    Balaban A.T.(ed), (1976) Chemical Applications of Graph Theory. Academic Press, LondonGoogle Scholar
  5. 5.
    Pólya G., Tarjan R.E., Woods D.R., (1983) Notes on Introductory Combinatorics. Birkhäuser, BostonGoogle Scholar
  6. 6.
    Balasubramanian K., (1985). Chem. Rev 85: 599–618CrossRefGoogle Scholar
  7. 7.
    E.K. Lloyd, in: Studies in Physical and Theoretical Chemistry. Graph Theory and Topology in Chemistry, vol.51 eds. R.B. King and D.H. Rouvray, (Elsevier, Amsterdam, 1987), pp. 537–543.Google Scholar
  8. 8.
    A.T. Balaban, in: Chemical Group Theory. Introduction and Fundamentals, eds. D. Bonchev and D.H. Rouvray (Gordon & Breach, Switzerland, 1994) pp. 159–208.Google Scholar
  9. 9.
    Fujita S., (2005) Theor. Chem. Acc. 113: 73–79CrossRefGoogle Scholar
  10. 10.
    Fujita S., (2005) Theor. Chem. Acc. 113: 80–86CrossRefGoogle Scholar
  11. 11.
    Fujita S., (1991) Symmetry and Combinatorial Enumeration in Chemistry. Springer-Verlag, Berlin-HeidelbergGoogle Scholar
  12. 12.
    Fujita S., (1995) Theor. Chem. Acta. 91: 291–314Google Scholar
  13. 13.
    Fujita S., (1995) Theor. Chem. Acta. 91: 315–332Google Scholar
  14. 14.
    Fujita S., (1998) Bull. Chem. Soc. Jpn. 71: 1587–1596CrossRefGoogle Scholar
  15. 15.
    Fujita S., (1999) Bull. Chem. Soc. Jpn. 72: 2409–2416CrossRefGoogle Scholar
  16. 16.
    Ruch E., Klein D.J., (1983) Theor. Chim. Acta. 63: 447–472Google Scholar
  17. 17.
    Brocas J., (1986) J. Amer. Chem. Soc. 108: 1135–1145CrossRefGoogle Scholar
  18. 18.
    Hässelbarth W., (1985) Theor. Chim. Acta. 67: 427–437CrossRefGoogle Scholar
  19. 19.
    Mead C.A., (1987) J. Amer. Chem. Soc. 109: 2130–2137CrossRefGoogle Scholar
  20. 20.
    Ruch E., Hässelbarth W., Richter B., (1970) Theor. Chim. Acta. 19: 288–300CrossRefGoogle Scholar
  21. 21.
    Fujita S., (1994) J. Graph Theor. 18: 349–371CrossRefGoogle Scholar
  22. 22.
    Fujita S., (1986) J. Chem. Educ 63: 744–746Google Scholar
  23. 23.
    Fujita S., (2002) Chem. Rec. 2: 164–176CrossRefGoogle Scholar
  24. 24.
    Fujita S., (2004) J. Math. Chem. 35: 261–283CrossRefGoogle Scholar
  25. 25.
    Fujita S., (1990) Bull. Chem. Soc. Jpn. 63: 203–215CrossRefGoogle Scholar
  26. 26.
    Fujita S., (1998) Theor. Chem. Acc. 99: 224–230CrossRefGoogle Scholar
  27. 27.
    Fujita S., (2000) J. Chem. Inf. Comput. Sci. 40: 1101–1112CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Chemistry and Materials TechnologyKyoto Institute of TechnologySakyokuJapan

Personalised recommendations