What question did this study set out to answer?

The aim is to develop sum rules for permutations of integers with specified fixed points using Stirling numbers.

May 20, 2026Open Access

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

Key Points

The aim is to develop sum rules for permutations of integers with specified fixed points using Stirling numbers.
Utilized partial sums of moments in permutations.
Employed Vassilev-Missana's formula and Schlomlich's expression of Stirling numbers.
Outlined relationships with Bell numbers.
Established identities involving Stirling numbers of the first kind.
Derived sum rules for binomial coefficients.
Clarified connections with Bell numbers.

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.

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