Cluster Algebras and Binary Subwords

Bailey, Rachel; Gunawan, Emily

doi:10.1007/s11083-021-09562-7

Cited by 2 publications

(7 citation statements)

References 16 publications

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“…The last term in the above sum n j=1 r j [2n − 2j + 1] q q j is [1] q q n , which shows that n j=1 r j [2n − 2j + 1] q q j has a maximum in degree n. That the last two terms are [1] q q n and 2 [3] q q n−1 implies that whatever the maximum coefficient is, the previous coefficient is one less. Now adding [n] q = [n − 1] q + q n−1 to the sum n j=1 r j q j [2n − 2j + 1] q and again considering the term [3] q q n−1 shows that the small plateau in question exists, in degrees n − 1 and n as claimed.…”

Section: Rank Unimodalitymentioning

confidence: 98%

“…1 shows one of the five lattice paths on the snake graph G bb dual to G ab .Note that the weight of this lattice path coincides with the weight of the the perfect matching shown in Figure4.1.…”

mentioning

confidence: 89%

“…The posets we define here are called piecewise-linear posets in [1], and zig-zag chains in [24]. with n elements and n − 1 edges.…”

Section: Posetsmentioning

confidence: 99%

“…That the maximum of L w (q) occurs in degree n+m 2 can be seen by computing the last term [1] q q n+m 2 in the sum.…”

Section: Rank Unimodalitymentioning

confidence: 99%

“…1 shows three snake graphs G w 1 , G w 2 , and G w 3 , along with their respective shapes and lattices. Below these figures, we indicate the respective rank generating functions.…”

mentioning

confidence: 99%

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Expansion Posets for Polygon Cluster Algebras

Claussen

2020

Preprint

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Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion formula. We introduce an involution on several of the interrelated combinatorial objects and constructions associated to type A surface cluster algebras, including certain classes of arcs, triangulations, and distributive lattices. We use these involutions to formulate a dual version of skein relations for arcs, and dual versions of three existing expansion posets. In particular, this leads to two new cluster expansion formulas, and recovers the lattice path expansion of Propp et al. We provide an explicit, structure-preserving poset isomorphism between an expansion poset and its dual version from the dual arc. We also show that an expansion poset and its dual version constructed from the same arc are dual in the sense of distributive lattices.We show that any expansion poset is isomorphic to a closed interval in one of the lattices L(m, n) of Young diagrams contained in an m × n grid, and that any L(m, n) has a covering by such intervals. In particular, this implies that any expansion poset is isomorphic to an interval in Young's lattice.We give two formulas for the rank function of any lattice path expansion poset, and prove that this rank function is unimodal whenever the underlying snake graph is built from at most four maximal straight segments. This gives a partial solution to a recent conjecture by Ovsienko and Morier-Genoud. We also characterize which expansion posets have symmetric rank generating functions, based on the shape of the underlying snake graph.

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Section: Rank Unimodalitymentioning

confidence: 98%

mentioning

confidence: 89%

“…The posets we define here are called piecewise-linear posets in [1], and zig-zag chains in [24]. with n elements and n − 1 edges.…”

Section: Posetsmentioning

confidence: 99%

“…That the maximum of L w (q) occurs in degree n+m 2 can be seen by computing the last term [1] q q n+m 2 in the sum.…”

Section: Rank Unimodalitymentioning

confidence: 99%

“…1 shows three snake graphs G w 1 , G w 2 , and G w 3 , along with their respective shapes and lattices. Below these figures, we indicate the respective rank generating functions.…”

mentioning

confidence: 99%

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Expansion Posets for Polygon Cluster Algebras

Claussen

2020

Preprint

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q-Rationals and Finite Schubert Varieties

Ovenhouse¹

2023

Comptes Rendus. Mathématique

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The classical q-analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give q-analogues of rational numbers. Some combinatorial interpretations are already known, namely as the rank generating functions for certain partially ordered sets. We give a new interpretation, showing that the numerators of q-rationals count the sizes of certain varieties over finite fields, which are unions of open Schubert cells in some Grassmannian.

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Cluster Algebras and Binary Subwords

Cited by 2 publications

References 16 publications

Expansion Posets for Polygon Cluster Algebras

Expansion Posets for Polygon Cluster Algebras

q-Rationals and Finite Schubert Varieties

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