On a conjecture of Lin and Kim concerning a refinement of Schr\"oder numbers

TL;DR

This paper confirms a recent conjecture by Lin and Kim by showing that the distribution of the first letter statistic on certain permutation avoidance classes matches entries in a new Schr"oder number triangle, using combinatorial techniques.

Contribution

It introduces a new Schr"oder number triangle and proves its connection to permutation avoidance classes, confirming Lin and Kim's conjecture.

Findings

Distribution matches entries of a new Schr"oder number triangle

Permutation classes share the same first letter statistic distribution

Employs generating trees, bijections, and kernel method in proofs

Abstract

In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schr\"oder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.