Flip-sort and combinatorial aspects of pop-stack sorting

Asinowski, Andrei; Banderier, Cyril; Hackl, Benjamin

doi:10.48550/arxiv.2003.04912

Cited by 9 publications

(18 citation statements)

References 14 publications

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“…The segment σ 3,6 (right) is the smallest segment contained in σ that includes all of the blocks in X 3,6 (σ). The violating pairs of T are (3,6) (because of the second row) and (5,6) (because of the first row). Since (3,6) is a violating pair of T , the segment σ 3,6 is forbidden.…”

Section: 2mentioning

confidence: 99%

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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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Section: 2mentioning

confidence: 99%

“…The authors of [5] suggested considering the average size of the forward orbit of a permutation in S n under the pop-stack-sorting map. In [25], the current author conjectured that this average number of iterations is asymptotically equal to n, which is the maximum possible size of a forward orbit by Ungar's theorem.…”

Section: Further Directionsmentioning

confidence: 99%

Pop-Stack-Sorting for Coxeter Groups

Defant¹

2021

Preprint

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“…We deduce that v can be written as shift β (ι(u)), where u = y 1 ⊕ • • • ⊕ y r ∈ S n and each y i is in Υ * κ i . Furthermore, there is a unique choice of β such that 0 ≤ β ≤ κ 1 − 1. hook H 1 has endpoints (−2, 0) and (2,3). The hook H 2 has endpoints (2, 3) and (4,6).…”

Section: Coxeter Stack-sorting In Type Amentioning

confidence: 99%

“…The second motivation for our terminology comes from the fact that when ≡ des is the descent congruence on W = S n , the map S ≡ des is the pop-stacksorting map. This function, which is a deterministic analogue of a pop-stack-sorting machine introduced by Avis and Newborn in [5], first appeared in a paper of Ungar's about directions determined by points in the plane [61]; it has received a great deal of attention over the past few years [1,3,4,18,19,26,32,47].…”

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confidence: 99%

Stack-Sorting for Coxeter Groups

Defant¹

2021

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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 Sn, the operator S≡ is West's stack-sorting map. When ≡ is the descent congruence on Sn, 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 Sn, then every permutation in the image of S≡ has at most ö 2(n−1) 3 ù 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 sB 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 sB 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 Sn.

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“…This implies that b j (µ) = 0 < r 0 for all 1 ≤ j ≤ f 0 . We also have b j (µ) ≤ r 0 for all 3,4); one can check that µ # is indeed a 2-Pop-sortable element of Tam(ν # ). Then (r 0 , .…”

mentioning

confidence: 93%

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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Flip-sort and combinatorial aspects of pop-stack sorting

Cited by 9 publications

References 14 publications

Pop-Stack-Sorting for Coxeter Groups

Pop-Stack-Sorting for Coxeter Groups

Stack-Sorting for Coxeter Groups

Meeting Covered Elements in $ν$-Tamari Lattices

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