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Three identities between Stirling numbers and the stabilizing character sequence


Author: Michael Gilpin
Journal: Proc. Amer. Math. Soc. 60 (1976), 360-364
MSC: Primary 05A15; Secondary 20B99
DOI: https://doi.org/10.1090/S0002-9939-1976-0414376-9
MathSciNet review: 0414376
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Abstract: Let $ \chi $ denote the stabilizing character of the action of the finite group G on the finite set X. Let $ {\chi _k}$ denote $ \vert G{\vert^{ - 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0414376-9
Keywords: Stirling numbers, orbit, stabilizing character, stabilizing subgroup
Article copyright: © Copyright 1976 American Mathematical Society

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