Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 1, 133-137

THE MOMENTS OF THE NUMBER OF CYCLES OF A RANDOM PERMUTATION BY SIMPLE ENUMERATION

By

T. LENGYEL, *Occidental College, Los Angeles*

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SUMMARY.We present a new proof for the Poisson limit distribution of the number of fixed points of a random permutation. Despite the combinatorial nature of the proof, it does not involve the use of inclusion-exclusion principle, cycle representation of permutations, number of derangements, rewriting formulas for the distribution of the fix points, generating functions or transformations formulas between moments. The proof is elementary in terms of enumeration and based on the notion of Stirling numbers. It requires some familiarity with the moment generating function of the Poisson distribution and the Frechet-Shohat moment convergence theorem. The method is extended to the distribution of the number of *k*-cycles of an *n*-element set, for *k*<= *n*

*AMS (1991) subject classification.* 60C05, 60E10, 60F05, 05A05, 05A19

*Key words and phrases*. Random permutation, moments, combinational probability.