On Simsun and Double Simsun Permutations Avoiding a Pattern of Length Three

TL;DR

This paper studies simsun and double simsun permutations avoiding length-three patterns, introduces a bijection with increasing 1-2 trees, and enumerates pattern-avoiding cases.

Contribution

It presents a new bijection between simsun permutations and increasing 1-2 trees, facilitating enumeration of pattern-avoiding double simsun permutations.

Findings

Established a bijection between simsun permutations and increasing 1-2 trees.

Enumerated double simsun permutations avoiding each pattern of length three.

Derived new enumeration formulas for pattern-avoiding simsun permutations.

Abstract

A permutation $\sigma\in\mathfrak{S}_n$ is simsun if for all $k$, the subword of $\sigma$ restricted to $\{1,...,k\}$ does not have three consecutive decreasing elements. The permutation $\sigma$ is double simsun if both $\sigma$ and $\sigma^{-1}$ are simsun. In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.