# Partial Rank Symmetry of Distributive Lattices for Fences

@article{Elizalde2022PartialRS, title={Partial Rank Symmetry of Distributive Lattices for Fences}, author={Sergi Elizalde and Bruce E. Sagan}, journal={Annals of Combinatorics}, year={2022} }

Associated with any composition β = (a, b, . . .) is a corresponding fence poset F (β) whose covering relations are x1 ✁ x2 ✁ . . .✁ xa+1 ✄ xa+2 ✄ . . .✄ xa+b+1 ✁ xa+b+2 ✁ . . . . The distributive lattice L(β) of all lower order ideals of F (β) is important in the theory of cluster algebras. In addition, its rank generating function r(q;β) is used to define q-analogues of rational numbers. Oğuz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a…

## One Citation

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## 33 References

### Rowmotion on fences

- 2021

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A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1<x_2<...<x_a>x_{a+1}>...>x_b<x_{b+1}<... where a, b, ... are positive integers. We investigate rowmotion on antichains and…

### Expansion Posets for Polygon Cluster Algebras

- 2020

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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…

### $h^\ast $-polynomials of zonotopes

- 2018

Mathematics

Transactions of the American Mathematical Society

The Ehrhart polynomial of a lattice polytope $P$ encodes information about the number of integer lattice points in positive integral dilates of $P$. The $h^\ast$-polynomial of $P$ is the numerator…

### Symmetric Decompositions and Real-Rootedness

- 2019

Mathematics

International Mathematics Research Notices

In algebraic, topological, and geometric combinatorics, inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently, a notion called the alternatingly…

### Dynamical algebraic combinatorics and the homomesy phenomenon

- 2016

Mathematics

We survey recent work within the area of algebraic combinatorics that has the flavor of discrete dynamical systems, with a particular focus on the homomesy phenomenon codified in 2013 by James Propp…

### Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a

- 1989

Mathematics

A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave…

### The Combinatorics of Frieze Patterns and Markoff Numbers

- 2020

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Integers

A matchings model is a combinatorial interpretation of Fomin and Zelevinsky's cluster algebras of type A that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns.

### Gamma-positivity in combinatorics and geometry

- 2017

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Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata,…