Enumerating permutations sortable by k passes through a pop-stack

Claesson, Anders; Guðmundsson, Bjarki Ágúst

doi:10.1016/j.aam.2019.04.002

Cited by 31 publications

(37 citation statements)

References 10 publications

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“…In particular, Pop ↓ L ( 0) = 0, and Pop ↑ L ( 1) = 1. As mentioned in the introduction, Pop ↓ L coincides with the classical pop-stack sorting map (see [CG19,EG21]) when L is the right weak order on the symmetric group S n . Pop-stack sorting operators on lattices were introduced in [Def21b, Def21a] as generalizations of the pop-stack sorting map.…”

Section: Pop-stack Sorting and Rowmotionmentioning

confidence: 79%

Semidistrim Lattices

Defant¹,

Williams²

2021

Preprint

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We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim, and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere.Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard's κ map on semidistributive lattices as well as Thomas and the second author's rowmotion on trim lattices. Every lattice has an associated pop-stack sorting operator that sends an element x to the meet of the elements covered by x. For semidistrim lattices, we are able to derive several intimate connections between rowmotion and pop-stack sorting, one of which involves independent dominating sets of the Galois graph.

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Section: Pop-stack Sorting and Rowmotionmentioning

confidence: 79%

Semidistrim Lattices

Defant¹,

Williams²

2021

Preprint

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“…Generating functions for 2-avoidance sets might also exhibit interesting behaviour. For the sets (F k , G k ) in Theorem 14 we know by [3] the generating functions are rational for all k, but for general sets F, G, the set Av 2 (F, G) could have interesting enumerations.…”

Section: Discussionmentioning

confidence: 99%

“…Pudwell and Smith characterised permutations sorted by 2 passes, in terms of avoiding a set of six usual patterns and two special barred patterns (defined in Subsection 2.1 below), and computed a rational generating function for the number of such permutations [11]. Claesson and Guðmundsson then computed a rational generating function for permutations sorted by any finite number of passes [3], and asked whether a "useful permutation pattern characterization of the k-pop stack-sortable permutations" exists for k 3.…”

Section: Introductionmentioning

confidence: 99%

$k$-Pop Stack Sortable Permutations and $2$-Avoidance

Elder

Goh

2021

Electron. J. Combin.

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We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb{N}$ the set is characterised by finitely many patterns, answering a question of Claesson and Guðmundsson. Moreover, these sets of patterns are algorithmically constructible. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called $2$-avoidance.

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“…As in the invertible case, it is natural to ask for the maximum size of a forward orbit; this question was explored for some specific noninvertible combinatorial dynamical systems in [2,10,23,26,27,29,33,35,44,47]. Another typical line of questions, especially in the case of sorting operators, asks for the characterization and/or enumeration of elements of X that require at most t iterations of f to reach a fixed point (see [1,13,20,22,26,28,31,39,48] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Recent work has focused on t-pop-stack-sortable permutations, which are permutations that require at most t iterations of the pop-stack-sorting map to reach the identity [2,20,28,39]. There has also been a great deal of work devoted to t-stack-sortable permutations in the study of West's stack-sorting map, especially for t ∈ {1, 2, 3} (see [1,13,22,25,31,48] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Meeting Covered Elements in $ν$-Tamari Lattices

Defant¹

2021

Preprint

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For each complete meet-semilattice M , we define an operator Pop M : M → M by Pop M (x) = ({y ∈ M : y x} ∪ {x}).When M is the right weak order on a symmetric group, Pop M is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of Pop Tam(ν) , where Tam(ν) is the ν-Tamari lattice. We determine the maximum size of a forward orbit of Pop Tam(ν) . When Tam(ν) is the n th m-Tamari lattice, this maximum forward orbit size is m + n − 1; in this case, we prove that the number of forward orbits of size m + n − 1 isMotivated by the recent investigation of the pop-stack-sorting map, we define a lattice path µ ∈ Tam(ν) to be t-Pop-sortable if Pop t Tam(ν) (µ) = ν. We enumerate 1-Pop-sortable lattice paths in Tam(ν) for arbitrary ν. We also give a recursive method to generate 2-Pop-sortable lattice paths in Tam(ν) for arbitrary ν; this allows us to enumerate 2-Pop-sortable lattice paths in a large variety of ν-Tamari lattices that includes the m-Tamari lattices.

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Enumerating permutations sortable by k passes through a pop-stack

Cited by 31 publications

References 10 publications

Semidistrim Lattices

Semidistrim Lattices

$k$-Pop Stack Sortable Permutations and $2$-Avoidance

Meeting Covered Elements in $ν$-Tamari Lattices

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