Related papers: Patterns in Shi tableaux and Dyck paths
We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply…
We introduce notions of linear reduction and linear equivalence of bijections for the purposes of study bijections between Young tableaux. Originating in Theoretical Computer Science, these notions allow us to give a unified view of a…
A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield to…
The Turing patterning mechanism is believed to underly the formation of repetitive structures in development, such as zebrafish stripes and mammalian digits, but it has proved difficult to isolate the specific biochemical species…
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets (which we rephrase as…
This paper considers a hyperplane arrangement constructed with a subset of a set of all simple paths in a graph. A connection of the constructed arrangement to the maximum matching problem is established. Moreover, the problem of finding…
In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in…
The theory of polyptych lattices is a framework to obtain a family of toric degenerations whose polytopes are related by piecewise-linear transformations. It can be regarded as a generalization of toric degenerations arising from cluster…
The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian…
Our main contribution here is the discovery of a new family of standard Young tableaux $ {\cal T}^k_n$ which are in bijection with the family ${\cal D}_{m,n}$ of Rational Dyck paths for $m=k\times n\pm 1$ (the so called "Fuss" case). Using…
We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$. Restricting our attention to when the rank $\delta$ is rational valued, we show that each countable model of the theory of a given Baldwin-Shi hypergraph is…
We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept…
Cracks in thin layers are influenced by what lies beneath them. From buried craters to crocodile skin, crack patterns are found over an enormous range of length scales. Regardless of absolute size, their substrates can dramatically…
Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented…
Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular…
Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational…
Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in $\Bbb{R}^n$, we introduce a cubic matrix for each region to establish two bijections in a uniform way. Firstly, the positions of minimal positive entries in column…
We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to…
How many matchings on the vertex set V={1,2,...,2n} avoid a given configuration of three edges? Chen, Deng and Du have shown that the number of matchings that avoid three nesting edges is equal to the number of matchings avoiding three…
Topological photonics provides a powerful framework to describe and understand many nontrivial wave phenomena in complex electromagnetic platforms. The topological index of a physical system is an abstract global property that depends on…