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.
Jean‐Christophe Pain (Sun,) studied this question.