Surprising symmetries in 132-avoiding permutations

TL;DR

This paper uncovers unexpected symmetries in the distribution of certain patterns within 132-avoiding permutations, revealing that different patterns occur equally often and generalizing this phenomenon to many pattern pairs.

Contribution

It proves a surprising symmetry in pattern counts in 132-avoiding permutations and extends this to a large class of pattern pairs with equal occurrence counts.

Findings

Equal pattern counts for 231, 312, 213 in 132-avoiding permutations

Combinatorial proof of the symmetry

Exponential number of pattern pairs with equal counts

Abstract

We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry. We then significantly generalize this result to show an exponential number of different pairs of patterns $q$ and $q'$ of length $k$ for which $S_{n,132}(q)=S_{n,132}(q')$ and the equality is non-trivial.