Three identities between Stirling numbers and the stabilizing character sequence

Gilpin, Michael

doi:10.1090/S0002-9939-1976-0414376-9

Three identities between Stirling numbers and the stabilizing character sequence
HTML articles powered by AMS MathViewer

by Michael Gilpin PDF

Proc. Amer. Math. Soc. 60 (1976), 360-364 Request permission

Abstract:

Let $\chi$ denote the stabilizing character of the action of the finite group G on the finite set X. Let ${\chi _k}$ denote $|G{|^{ - 1}}{\Sigma _{\sigma \in G}}\chi {(\sigma )^k}$ It is well known that ${\chi _k}$ is the number of orbits of the induced action of G on the Cartesian product ${X^{(k)}}$. We show if G is a least $(k - 1)$-fold transitive on X, then ${\chi _k}$ can be expressed in terms of Stirling numbers of both kinds. Three identities between Stirling numbers and the stabilizing character sequence are obtained.

References

W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 0069818
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
H. W. Gould, Stirling number representation problems, Proc. Amer. Math. Soc. 11 (1960), 447–451. MR 114767, DOI 10.1090/S0002-9939-1960-0114767-8
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
C. L. Liu, Introduction to combinatorial mathematics, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR 0234840
Donald Passman, Permutation groups, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0237627

Kombinatorishe Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen

68

John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
Louis W. Shapiro, Finite Groups Acting on Sets with Applications, Math. Mag. 46 (1973), no. 3, 136–147. MR 1572044