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

Pain, Jean-Christophe (Paris-Saclay University)

Indonesian Journal of Combinatorics
Vol 9, No 2 (2025)

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.