Hopping from Chebyshev polynomials to permutation statistics

TL;DR

This paper establishes formulas linking permutation statistics with Chebyshev polynomial generating functions, using combinatorial methods like tilings and valley-hopping, and explores cyclic analogues for derangements.

Contribution

It introduces new formulas connecting permutation statistics to Chebyshev polynomials and their cyclic analogues, with combinatorial proofs and applications.

Findings

Formulas for permutation statistics via Chebyshev polynomials

Cyclic analogues of these formulas for derangements

Applications to $(-1)$-evaluations of distributions

Abstract

We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the $(-1)$-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic analogue of this action due to Sun and Wang.