On Two Bijections from S_n(321) to S_n(132)

TL;DR

This paper compares two bijections between 321-avoiding and 132-avoiding permutations, revealing their simple relationship through pictorial reformulations and symmetry operations.

Contribution

It provides a pictorial reformulation of Elizalde and Pak's bijection, showing its relation to Bloom and Saracino's bijection via inversion, reversal, and complementation.

Findings

The two bijections are related by simple symmetry operations.

Pictorial reformulations clarify the relationship between the bijections.

Both bijections preserve fixed points and excedances.

Abstract

Let S_n(321) (respectively, S_n(132)) denote the set of all permutations of {1,2,...,n} that avoid the pattern 321 (respectively, the pattern 132). Elizalde and Pak gave a bijection Theta from S_n(321) to S_n(132) that preserves the numbers of fixed points and excedances for each element of S_n(321), and commutes with the operation of taking inverses. Bloom and Saracino proved that another bijection Gamma from S_n(321) to S_n(132), introduced by Robertson, has the same properties, and they later gave a pictorial reformulation of Gamma that made these results more transparent. Here we give a pictorial reformulation of Theta, from which it follows that, although the original definitions of Theta and Gamma are very different, these two bijections are in fact related to each other in a very simple way, by using inversion, reversal, and complementation.