The pop-stack-sorting operator on Tamari lattices

Hong, Letong

doi:10.1016/j.aam.2022.102362

Cited by 9 publications

(5 citation statements)

References 7 publications

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“…In the time since we released the initial preprint of this article, several parts of Theorem 11.2 have been resolved. The part of the conjecture concerning was settled by Hong in [27]. The parts of the conjecture concerning , , and were settled by Choi and Sun in [11].…”

Section: Further Directionsmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

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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 $\overline \kappa $ 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: Further Directionsmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

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“…For the proof of Lemma 3.9, we make use of the following theorem by Hong [10]. A double descent of a permutation is a 321 pattern.…”

Section: Weak(b Nmentioning

confidence: 99%

“…We now proceed to the proof of Theorem 1.2, in which we make use of the following result by Hong [10].…”

Section: We Arrive Atmentioning

confidence: 99%

“…In this paper, we resolved the conjectures proposed by Defant and Williams [8] concerning Pop(M; q) when M is Weak(B n ), Tam(B n ), J(Φ + (A n )), or J(Φ + (B n )). Previously, their conjecture concerning Pop(M; q) for M = Tam(A n ) was proven by Hong [10]. The last of their six conjectures remains open; it concerns a formula for Pop(M; q) when M = Camb bi (A n ) is the Cambrian lattice of type A n arising from a bipartite Coxeter element.…”

Section: Future Directionsmentioning

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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“…The articles [22,28,29,32,34,47] study the dynamical properties and the image of Pop L for various choices of interesting lattices L.…”

Section: Introductionmentioning

confidence: 99%

Ungarian Markov Chains

Defant¹,

Li²

2023

Preprint

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We introduce the Ungarian Markov chain UL associated to a finite lattice L. The states of this Markov chain are the elements of L. When the chain is in a state x ∈ L, it transitions to the meet of {x} ∪ T , where T is a random subset of the set of elements covered by x. We focus on estimating E(L), the expected number of steps of UL needed to get from the top element of L to the bottom element of L. Using direct combinatorial arguments, we provide asymptotic estimates when L is the weak order on the symmetric group Sn and when L is the n-th Tamari lattice. When L is distributive, the Markov chain UL is equivalent to an instance of the well-studied random process known as last-passage percolation with geometric weights. One of our main results states that if L is a trim lattice, then E(L) ≤ E(spine(L)), where spine(L) is a specific distributive sublattice of L called the spine of L. Combining this lattice-theoretic theorem with known results about last-passage percolation yields a powerful method for proving upper bounds for E(L) when L is trim. We apply this method to obtain uniform asymptotic upper bounds for the expected number of steps in the Ungarian Markov chains of Cambrian lattices of classical types and the Ungarian Markov chains of ν-Tamari lattices.

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The pop-stack-sorting operator on Tamari lattices

Cited by 9 publications

References 7 publications

Semidistrim Lattices

Semidistrim Lattices

The Image of the Pop Operator on Various Lattices

Ungarian Markov Chains

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