Sum rules for permutations with fixed points involving Stirling numbers of the first kind - Jean-Christophe Pain (Indonesian Combinatorial Society (InaCombS), 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.