Determinantal representations of alternating run polynomials

TL;DR

This paper derives determinantal formulas for various alternating run polynomials, revealing new recurrence relations and connections to Eulerian and Stirling permutation polynomials.

Contribution

It introduces novel determinantal representations for Eulerian, Andre, and alternating run polynomials, including their relations to Stirling permutations and type B Eulerian polynomials.

Findings

Euler number expressed as a lower Hessenberg determinant

New recurrence relations for alternating run polynomials

Connection between Stirling permutation polynomials and type B Eulerian polynomials

Abstract

Based on a determinantal formula for the higher derivative of a quotient of two functions, we first present the determinantal expressions of Eulerian polynomials and Andre polynomials. In particular, we discover that the Euler number (number of alternating permutations) can be expressed as a lower Hessenberg determinant. We then investigate the determinantal representations of the up-down run polynomials and the types A and B alternating run polynomials. As applications, we deduce several new recurrence relations, which imply the multiplicity of -1 in these three kinds of polynomials. And then, we provide two determinantal representations for the alternating run polynomials of dual Stirling permutations. In particular, we discover a close connection between the alternating run polynomials of dual Stirling permutations and the type B Eulerian polynomials.