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

Elder, Murray; Goh, Yoong Kuan

doi:10.37236/9606

Cited by 11 publications

(14 citation statements)

References 6 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: 82%

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: 82%

Semidistrim Lattices

Defant¹,

Williams²

2021

Preprint

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“…In the case of West's stack-sorting map, these elements are the t-stack-sortable permutations, which have been studied extensively, especially for t ∈ {1, 2, 3} (see [3,13,14,16,24,29,30] and the references therein). For the pop-stack-sorting map, these elements are the t-pop-stack-sortable permutations investigated in [22,34,52]. Given a Coxeter group W , let us say an element w ∈ W is t-pop-stack-sortable if Pop t W (w) = e. In other words, the set of t-pop-stack-sortable elements of W is Pop −t W (e).…”

mentioning

confidence: 99%

Pop-Stack-Sorting for Coxeter Groups

Defant¹

2021

Preprint

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Let W be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator Pop : W → W to be the map that fixes the identity element and sends each nonidentity element w to the meet of the elements covered by w in the right weak order. When W is the symmetric group Sn, Pop coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that sup w∈W |O Pop (w)| = h, where O Pop (w) is the forward orbit of w under Pop and h is the Coxeter number of W (with h = ∞ if W is infinite). More generally, we define a map f : W → W to be compulsive if for every w ∈ W , f (w) is less than or equal to Pop(w) in the right weak order. We prove that if f is compulsive, then supThis result is new even for symmetric groups.We prove that 2-pop-stack-sortable elements in type B are in bijection with 2-pop-stack-sortable permutations in type A, which were enumerated by Pudwell and Smith. Claesson and Guðmundsson proved that for each fixed nonnegative integer t, the generating function that counts t-pop-stacksortable permutations in type A is rational; we establish analogous results in types B and A.

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

Cited by 11 publications

References 6 publications

Semidistrim Lattices

Semidistrim Lattices

Pop-Stack-Sorting for Coxeter Groups

Meeting Covered Elements in $ν$-Tamari Lattices

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