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

Jean-Christophe Pain

Abstract

We propose sum rules for permutations $p_n(k)$ of the ensemble $\left\{1,2,\cdots,n\right\}$ 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 Schlömlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers $B_n$ are outlined.

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

Abstract

We propose sum rules for permutations of the ensemble with fixed points, in the form of partial sums of their moments. The corresponding identities involve Stirling numbers of the first kind . Using a formula due to Vassilev-Missana and the Schlömlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers are outlined.
Paper Structure (9 sections, 1 theorem, 65 equations)

This paper contains 9 sections, 1 theorem, 65 equations.

Table of Contents

  1. Introduction
  2. Sum rules involving Stirling numbers of the first kind
  3. The generating-function technique
  4. Extracting the coefficients
  5. Binomial sum rules
  6. Conclusion
  7. Bounds and asymptotic forms
  8. Properties of Stirling numbers of the first kind
  9. Proof of the binomial expression of $k^i$

Key Result

Theorem 1

For $n$ a non-zero natural number, and $m$ a natural number, one has where $p_n(k)$ is the permutation of $\left\{1,2\cdots,n\right\}$ with $k$ fixed points and $s(q,r)$ represents the (signed) Stirling number of the first kind.

Theorems & Definitions (2)

  • Theorem 1
  • proof