Restricted permutations and Chebyshev polynomials

TL;DR

This paper investigates generating functions for permutations with two restrictions, revealing that many cases can be expressed using Chebyshev polynomials of the second kind, thus connecting permutation enumeration with classical orthogonal polynomials.

Contribution

It introduces a novel connection between restricted permutation enumeration and Chebyshev polynomials, expanding the understanding of permutation generating functions.

Findings

Many restricted permutation generating functions are expressible via Chebyshev polynomials

The restrictions involve permutations in  and , with results extending to  and  cases

The approach unifies various cases under a Chebyshev polynomial framework

Abstract

We study generating functions for the number of permutations in $\SS_n$ subject to two restrictions. One of the restrictions belongs to $\SS_3$, while the other to $\SS_k$. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind.