The classification of 231-avoiding permutations by descents and maximum drop

TL;DR

This paper investigates 231-avoiding permutations with constraints on descents and maximum drop, establishing connections to Dyck paths and ordered trees, and deriving explicit formulas and recursions for their enumeration.

Contribution

It introduces new combinatorial interpretations and explicit formulas for counting 231-avoiding permutations with bounded descents and drops, linking them to Dyck paths and ordered trees.

Findings

Derived generating functions satisfying simple recursions.

Established bijections with Dyck paths and ordered trees.

Provided explicit formulas for special cases.

Abstract

We study the number of 231-avoiding permutations with $j$-descents and maximum drop is less than or equal to $k$ which we denote by $a_{n,231,j}^{(k)}$. We show that $a_{n,231,j}^{(k)}$ also counts the number of Dyck paths of length $2n$ with $n-j$ peaks and height $\leq k+1$, and the number of ordered trees with $n$ edges, $j+1$ internal nodes, and of height $\leq k+1$. We show that the generating functions for the $a_{n,231,j}^{(k)}$s with $k$ fixed satisfy a simple recursion. We also use the combinatorics of ordered trees to prove new explicit formulas for $a_{n,231,j}^{(k)}$ as a function of $n$ in a number of special values of $j$ and $k$ and prove a simple recursion for the $a_{n,231,j}^{(k)}$s.