Abstract
We study sorting operators $\textrm{A}$ on permutations that are obtained composing Knuth's stack sorting operator \textrmS and the reverse operator $\textrm{R}$, as many times as desired. For any such operator $\textrm{A}$, we provide a bijection between the set of permutations sorted by $\textrm{S} \circ \textrm{A}$ and the set of those sorted by $\textrm{S} \circ \textrm{R} \circ \textrm{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on an apparently novel bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding families of Wilf-equivalent permutation classes.
Highlights
Partial sorting algorithms were one of the early motivations for the study of permutation patterns
We will prove that ΦA is a bijection from the set of permutations sorted by S ◦ A to the set of those sorted by S ◦ R ◦ A
We study the properties of bijections ΦA in somewhat greater detail. This will prove the second part of Conjecture 1, that deals with permutation statistics equidistributed over the set of permutations sorted by S ◦ A and the set of those sorted by S ◦ R ◦ A
Summary
Partial sorting algorithms were one of the early motivations for the study of permutation patterns. Stack sorting can be considered as an operator or procedure, S, applied to permutations It is defined recursively as: S(αnβ) = S(α)S(β)n. Stack sortable permutations are those that may not contain subwords (not necessarily consecutive) of the form bca where a < b < c Such permutations are said to avoid the pattern 231, and the collection of all such is denoted Av(231). We can replace this latter set by the elements of Av(132) belonging to the image of A, since the self-inverse operator R immediately provides a bijection between Av(231) and Av(132) In establishing this result we demonstrate an apparently novel bijection between Av(231) and Av(132) which preserves many permutation statistics.
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