A Simple Bijection Between $231$-Avoiding and $312$-Avoiding Placements

Abstract

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