Lagrange Inversion Counts $3\bar{5}241$-Avoiding Permutations

TL;DR

This paper presents a simplified proof using Lagrange inversion to count permutations avoiding the pattern 3̅5241, building on prior work that involved a complex bijection.

Contribution

The authors provide a more straightforward proof for counting 3̅5241-avoiding permutations using Lagrange inversion, replacing a complicated bijection from previous research.

Findings

Counting sequence starts with 1 and is fixed under self-composition

Simpler proof method established for pattern-avoiding permutations

Connects permutation enumeration with Lagrange inversion techniques

Abstract

In a previous paper, we showed that $3\bar{5}241$-avoiding permutations are counted by the unique sequence that starts with a 1 and shifts left under the self-composition transform. The proof uses a complicated bijection. Here we give a much simpler proof based on Lagrange inversion.