Sum rules for permutations with fixed points involving Stirling numbers of the first kind

Jean-Christophe Pain

Abstract


We propose sum rules for permutations pn(k) of the ensemble {1,2,...,n} with k fixed points, in the form of partial sums of their moments. The corresponding identities involve Stirling numbers of the first kind s(q,r). Using a formula due to Vassilev-Missana and the Schlomlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers Bn are outlined.

Keywords


partitions; fixed points; Bell numbers; signed and unsigned Stirling numbers of the first kind; Stirling numbers of the second kind; binomial sums

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DOI: http://dx.doi.org/10.19184/ijc.2025.9.2.5

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