Catalan and Schröder permutations sortable by two restricted stacks

Baril, Jean-Luc; Cerbai, Giulio; Khalil, Carine; Vajnovszki, Vincent

doi:10.1016/j.ipl.2021.106138

Cited by 6 publications

(5 citation statements)

References 13 publications

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“…Remark 1.3. The authors of the recent paper [15] introduced a different generalization of West's stack-sorting map that uses pattern-avoiding stacks; this notion has spawned several subsequent articles in recent years [6,7,14,16,30]. While these pattern-avoiding stacks are certainly interesting, we believe our Coxeter stack-sorting operators are more natural from an algebraic and lattice-theoretic point of view.…”

Section: Introduction 1coxeter Stack-sorting Operatorsmentioning

confidence: 99%

“…Our construction guarantees that s(w) = P(T ) = v. It follows from the preceding paragraph that every preimage of v under s arises uniquely in this way. (4,6), and H 2 has southwest endpoint (4,6) and northeast endpoint (5,7). Then z 1 = (−1)(−2)23, and z 2 is the empty permutation.…”

mentioning

confidence: 99%

“…Figure 8.3. For example, H 1 has southwest endpoint (−1, 1) and northeast endpoint(4,6), and H 2 has southwest endpoint(4,6) and northeast endpoint(5,7). Then z 1 = (−1)(−2)23, and z 2 is the empty permutation.…”

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confidence: 99%

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Stack-sorting for Coxeter groups

Defant¹

2022

Combinatorial Theory

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Given an essential semilattice congruence ≡ on the left weak order of a Coxeter group W , we define the Coxeter stack-sorting operator S ≡ : W → W by S ≡ (w) = w π ≡ ↓ (w) −1 , where π ≡ ↓ (w) is the unique minimal element of the congruence class of ≡ containing w. When ≡ is the sylvester congruence on the symmetric group S n , the operator S ≡ is West's stack-sorting map. When ≡ is the descent congruence on S n , the operator S ≡ is the pop-stack-sorting map. We establish several general results about Coxeter stack-sorting operators, especially those acting on symmetric groups. For example, we prove that if ≡ is an essential lattice congruence on S n , then every permutation in the image of S ≡ has at most 2(n−1) /3 right descents; we also show that this bound is tight.We then introduce analogues of permutree congruences in types B and A and use them to isolate Coxeter stack-sorting operators s B and s that serve as canonical type-B and type-A counterparts of West's stack-sorting map. We prove analogues of many known results about West's stack-sorting map for the new operators s B and s. For example, in type A, we obtain an analogue of Zeilberger's classical formula for the number of 2-stack-sortable permutations in S n .

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Section: Introduction 1coxeter Stack-sorting Operatorsmentioning

confidence: 99%

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confidence: 99%

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Stack-sorting for Coxeter groups

Defant¹

2022

Combinatorial Theory

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“…The first two [11,12] were devoted to the study of the so-called σ-sortable permutations, for specific choices of σ. The (σ, τ )-machine [1,2] generalized the σ-machine to pairs of patterns. The current author [8] extended the domain of σ-machines to Cayley permutations.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we analyze some dynamical aspects of σ-machines by regarding a σ-stack as an operator π → S σ (π). This approach has been adopted recently in several works related to stack sorting [1,2,8,14]. Here we introduce some new properties and provide the first related results.…”

Section: Introductionmentioning

confidence: 99%

Dynamical aspects of $σ$-machines

Cerbai¹

2022

Preprint

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The σ-machine was recently introduced by Cerbai, Claesson and Ferrari as a tool to gain a better insight on the problem of sorting permutations with two stacks in series. It consists of two consecutive stacks, which are restricted in the sense that their content must at all times avoid a certain pattern: a given σ, in the first stack, and 21, in the second. Here we prove that in most cases sortable permutations avoid a bivincular pattern ξ. We provide a geometric decomposition of ξ-avoiding permutations and use it to count them directly. Then we characterize the permutations with the property that the output of the σ-avoiding stack does not contain σ, which we call effective. For σ = 123, we obtain an alternative method to enumerate sortable permutations. Finally, we classify σ-machines and determine the most challenging to be studied.

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