Classification of bijections between 321- and 132-avoiding permutations

TL;DR

This paper systematically analyzes and classifies all known bijections between 321- and 132-avoiding permutations, revealing their interrelations and extending the understanding of the statistics they preserve.

Contribution

It provides a comprehensive survey of bijections between these permutations, classifies them by preserved statistics, and offers a recursive description of a key bijection.

Findings

All known bijections are related via trivial transformations.

The classification extends known results on preserved statistics.

A recursive algorithmic description of a prominent bijection is provided.

Abstract

It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and show how they are related to each other via ``trivial'' bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial extensions of known results. Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics--the largest number of statistics any of the bijections respects.