相关论文: A bijection between dominant Shi regions and core …
Motivated by the relation holding for the m-generalized Catalan numbers of type A and C, the connection between dominant regions of the m-Shi arrangement of type A and C is investigated. In the same line of thought, a bijection between mn+1…
We extend the bijection of Fishel-Vazirani on dominant regions of the $m$-Shi arrangement. Our map puts the set of all minimal chambers of the $m$-Shi arrangement of Type $A_{n}$ in bijection with a certain set of (equivalence classes of)…
An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus…
The number of regions of the type A_{n-1} Shi arrangement in R^n is counted by the intrinsically beautiful formula (n+1)^{n-1}. First proved by Shi, this result motivated Pak and Stanley as well as Athanasiadis and Linusson to provide…
In this paper we present a bijection $\omega_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement…
In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and…
Given a Shi arrangement $\mathcal{A}_\Phi$, it is well-known that the total number of regions is counted by the parking number of type $\Phi$ and the total number of regions in the dominant cone is given by the Catalan number of type…
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…
The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved by…
We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we…
The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{n+1}\binom{2n}{n}$ when $\Psi=A_{n-1}$, and the binomial $\binom{2n}{n}$ when $\Psi=B_n$, and these numbers coincide with the correspondent…
In this paper, we give a bijection between rooted labeled ordered forests with a selected subset of their leaves and the regions of the type $C$ Catalan arrangement in $\R^n$. We thus obtain a bijective proof of the well-known enumeration…
Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check{Q}$, spanning a Euclidean space $V$. Let $m$ be a positive integer and $\aA^m_\Phi$ be the arrangement of hyperplanes in $V$ of the…
We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_i-x\_j=s$ for some integer $s$. Classical…
The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\Phi$ is a well-studied object enumerated by the Fu{\ss}-Catalan number $Cat^{(k)}(\Phi)$. It is natural to refine this enumeration by considering…
We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some…
Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper…
The notion of $(a,b)$-cores is closely related to rational $(a,b)$ Dyck paths due to Anderson's bijection, and thus the number of $(a,a+1)$-cores is given by the Catalan number $C_a$. Recent research shows that $(a,a+1)$ cores with distinct…
Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural…
This note introduces some bijections relating core partitions and tuples of integers. We apply these bijections to count the number of cores with various types of restriction, including fixed number of parts, limited size of parts, parts…