Counting pop-stacked permutations in polynomial time

Claesson, Anders; Guðmundsson, Bjarki Ágúst; Pantone, Jay

doi:10.48550/arxiv.1908.08910

Cited by 5 publications

(7 citation statements)

References 2 publications

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“…A descent of a permutation x = x 1 x 2 • • • x n ∈ S n is an adjacent pair of numbers x i and x i+1 such that x i > x i+1 . For x = 51763284, the descents of x are (5, 1), (7,6), (6,3), (3,2), and (8,4). We let |des(x)| represent the number of descents in x.…”

Section: Preliminariesmentioning

confidence: 99%

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The Image of the Pop Operator on Various Lattices

Choi¹,

Sun²

2022

Preprint

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Extending the classical pop-stack sorting map on the lattice given by the right weak order on S n , Defant defined, for any lattice M , a map Pop M : M → M that sends an element x ∈ M to the meet of x and the elements covered by x. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study Pop M (M ) when M is the weak order of type B n , the Tamari lattice of type B n , the lattice of order ideals of the root poset of type A n , and the lattice of order ideals of the root poset of type B n . In particular, we settle four conjectures proposed by Defant and Williams on the generating functionwhere U M (b) is the set of elements of M that cover b.

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Section: Preliminariesmentioning

confidence: 99%

“…The analog of Pop(M; q) for the pop-stack sorting map was previously studied [1,3]. Notably, in Asinowski, Banderier, Billey, Hackl, and Linusson [1], it was proved that [q n−2 ]Pop(Weak(A n−1 ); q) = 2 n − 2n, where Weak(A n−1 ) is the weak order on the symmetric group S n .…”

Section: Introductionmentioning

confidence: 99%

The Image of the Pop Operator on Various Lattices

Choi¹,

Sun²

2022

Preprint

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“…A great amount of research in combinatorics and computer science has focused on sorting operators, which are dynamical systems on S n that have the identity permutation e = 123 • • • n as their unique periodic (necessarily fixed) point. Some typical examples of such operators include the bubble sort map (see [17,2] and [37, pages 106-110]), West's stack-sorting map (see [10,21,22,26,53] and the references therein), the map revstack defined in [28], the pop-stack-sorting map [4,5,6,29,19,20,45,52], and the Queuesort map [18,40].…”

Section: Introduction 1sorting Operatorsmentioning

confidence: 99%

Pop-stack-sorting for Coxeter groups

Defant¹

2022

Combinatorial Theory

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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 S n , Pop coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove thatwhere h is the Coxeter number of W (with h = ∞ if W is infinite) and O f (w) denotes the forward orbit of w under a map f . When W is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of W is h−1. 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 sup w∈W |O f (w)| ⩽ h. This result is new even for symmetric groups.We prove that 2-pop-stack-sortable elements in type B are in bijection with 2-pop-stacksortable 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-stack-sortable permutations in type A is rational; we establish analogous results in types B and A.

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“…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

Preprint

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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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Counting pop-stacked permutations in polynomial time

Cited by 5 publications

References 2 publications

The Image of the Pop Operator on Various Lattices

The Image of the Pop Operator on Various Lattices

Pop-stack-sorting for Coxeter groups

Stack-Sorting for Coxeter Groups

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