A Simple Bijective Proof of the Shape-Wilf-Equivalence of the Patterns 231 and 312

TL;DR

This paper provides a straightforward bijective proof demonstrating the shape-Wilf-equivalence of the permutation patterns 231 and 312, simplifying previous complex proofs and offering new insights into pattern avoidance.

Contribution

It introduces a novel characterization of avoiding full rook placements, enabling a simple bijective proof of the shape-Wilf-equivalence between 231 and 312.

Findings

Established a bijective proof of shape-Wilf-equivalence

Simplified understanding of pattern avoidance for 231 and 312

Provided new characterization of rook placements avoiding these patterns

Abstract

Stankova and West proved in 2002 that the patterns 231 and 312 are shape-Wilf-equivalent. Their proof was nonbijective and fairly complicated. We give a new characterization of 231 and 312 avoiding full rook placements and use this to give a simple bijective proof of the shape-Wilf- equivalence.