Classification of bijections between 321- and 132-avoiding permutations

Anders Claesson,Sergey Kitaev

doi:10.46298/dmtcs.3594

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 confirming 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 we show how they are related to each other (via "trivial'' bijections). Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also analyze how many permutation statistics (from a fixed, but large, set of statistics) each of the known bijections preserves, obtaining substantial extensions of known results. 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 respect).

Highlights

Given two different bijections between two sets of combinatorial objects, what does it mean to say that one bijection is better than the other? Perhaps, a reasonable answer would be “The one that is easier to describe.” While the ease of description and how easy it is to prove properties of the bijection using the description is one aspect to consider, an even more important aspect, in our opinion, is how well the bijection reflects and translates properties of elements of the respective sets.A natural measure for a bijection between two sets of permutations, is how many statistics the bijection preserves
We don’t have an exhaustive list of permutation statistics, but we have used the following list as our “base” set: asc, des, exc, ldr, rdr, lir, rir, zeil, comp, lmax, lmin, rmax, rmin, head, last, peak, valley, lds, lis, rank, cyc, fp, slmax
To make sure we find all statistics that a given bijection “essentially”

Summary

Introduction

Given two different bijections between two sets of combinatorial objects, what does it mean to say that one bijection is better than the other? Perhaps, a reasonable answer would be “The one that is easier to describe.” While the ease of description and how easy it is to prove properties of the bijection using the description is one aspect to consider, an even more important aspect, in our opinion, is how well the bijection reflects and translates properties of elements of the respective sets. Knuth (6; 7) showed that the number of permutations avoiding a pattern of length 3 is independent of the pattern To prove this it suffices, due to symmetry, to consider one representative from {123, 321} and one from {132, 231, 213, 312}. The symmetry means that to prove this bijectively, it suffices to find a bijection from the set of permutations avoiding a pattern in one of the classes to permutations avoiding a pattern in the other. Theorem 2 For bijections from 321- to 132-avoiding permutations we have the following equidistribution results These results are maximal in the sense that they cannot be non-trivially extended by other statistics from STAT.

Permutation statistics

Bijections in the literature

A recursive description of Knuth-Richards’ bijection