Restricted 132-Involutions and Chebyshev Polynomials

TL;DR

This paper investigates the enumeration of involutions avoiding certain patterns, revealing that generating functions often relate to Chebyshev polynomials and establishing equivalences between different pattern-avoidance classes.

Contribution

It introduces new connections between involution pattern avoidance and Chebyshev polynomials, providing explicit generating functions and combinatorial interpretations.

Findings

Generating functions depend only on pattern length and relate to Chebyshev polynomials.

Involutions avoiding 132 and increasing patterns share the same enumeration as those avoiding double-wedge patterns.

Many results are supported by combinatorial proofs.

Abstract

We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $\tau$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and $12... k$ have the same enumerative formula according to the length than involutions avoiding both 132 and any {\em double-wedge pattern} possibly followed by fixed points of total length $k$. Many results are also shown with a combinatorial point of view.