Alternating runs of permutations and the central factorial numbers

TL;DR

This paper explores the relationships between the number of permutations with a given number of alternating runs and central factorial numbers, providing explicit formulas for related polynomials and functions.

Contribution

It establishes new connections between alternating run counts in permutations and central factorial numbers, including explicit formulas for peak and derivative polynomials.

Findings

Relationships between R(n,k) and central factorial numbers of even and odd indices.

Explicit formulas for peak and left peak polynomials.

Derivation of derivative polynomials of tangent and secant functions.

Abstract

Let R(n,k) be the number of permutations of $\{1,2,\ldots,n\}$ with k alternating runs. In this paper, we establish the relationships between R(n,k) and the central factorial numbers of even indices as well as the number of signed permutations with a given number of alternating runs and the central factorial numbers of odd indices. The explicit formulas of the peak and left peak polynomials for permutations and the derivative polynomials of the tangent and secant functions are also established.