Classification of bijections between 321- and 132-avoiding permutations

Abstract

International audience 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).

Fertility, Strong Fertility, and Postorder Wilf Equivalence

“…Assume by way of contradiction that swu(π) contains 3241. Using (14) and the fact that swu(R) and swu(L) avoid 3241, we deduce that swu(L) has a descent and swu(R) is nonempty. This implies that L has a descent and R is nonempty.…”

“…an entry π i such that π j < π i whenever i+1 ≤ j ≤ n. Let lmax(π) and rmax(π) denote the number of left-to-right maxima of π and the number of right-to-left maxima of π, respectively. The Zeilberger statistic, which originated in Zeilberger's study of 2-stack-sortable permutations [42] and has received attention in subsequent articles such as [7,9,14,22], is denoted by zeil. For π ∈ S n , zeil(π) is defined to be the largest integer m such that the entries n, n − 1, .…”

“…Let LenDes denote the set of permutation statistics f such that f (π) only depends on the length of π and the descent set Des(π). The set of statistics discussed in [9] (see [9,14] for all of their definitions) is LenDes ∪ {lmax, rmax, zeil, indmax, slmax, slmax • rev}.…”

“…π = L ⊕ (1 R) and σ = ( L ⊕ 1) R (13) so that swu(π) = (swu(L) ⊕ 1) swu(R) and swu −1 (σ) = swu −1 ( L) ⊕ (1 swu −1 ( R)). (14) Because L, R ∈ Av(231, µ), it follows by induction that swu(L) and swu(R) avoid swu(µ). A similar argument shows that swu −1 ( L) and swu −1 ( R) avoid µ.…”

“…Now assume swu −1 (σ) contains 1423. Because swu −1 ( R) avoids 1423, we can use (14) to see that swu −1 ( L) is nonempty. Similarly, swu −1 ( R) has an ascent because swu −1 ( L) avoids 1423.…”