The Image of the Pop Operator on Various Lattices

Choi, Yunseo; Sun, Nathan

doi:10.48550/arxiv.2209.13695

Cited by 2 publications

(2 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The part of the conjecture concerning Tamari( 𝐴 𝑛 ) was settled by Hong in [27]. The parts of the conjecture concerning Weak(𝐵 𝑛 ), Tamari(𝐵 𝑛 ), 𝐽 (Φ + 𝐴 𝑛 ) and 𝐽 (Φ + 𝐵 𝑛 ) were settled by Choi and Sun in [11].…”

Section: Enumerationmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

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.

show abstract

Section: Enumerationmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

The Image of the Pop Operator on Various Lattices

Cited by 2 publications

References 11 publications

Semidistrim Lattices

Semidistrim Lattices

Ungarian Markov Chains

Contact Info

Product

Resources

About