Related papers: From m-clusters to m-noncrossing partitions via ex…

We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of…

The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum_{u,w\in P} ^{}\mu(u,w) x^{\rk u}y^{\rk w}$, where $\mu(.,.)$ is the M\"obius function of $P$. We compute the $M$-triangle of Armstrong's poset of…

A partition on $[n]$ has a crossing if there exists $i\_1<i\_2<j\_1<j\_2$ such that $i\_1$ and $j\_1$ are in the same block, $i\_2$ and $j\_2$ are in the same block, but $i\_1$ and $i\_2$ are not in the same block. Recently, Chen et al.…

We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing…

We interpret noncrossing partitions of type $B$ and type $D$ in terms of noncrossing partitions of type $A$. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type $B$, type $C$ and type…

In this paper, we present a reduction algorithm which transforms $m$-regular partitions of $[n]=\{1, 2, ..., n\}$ to $(m-1)$-regular partitions of $[n-1]$. We show that this algorithm preserves the noncrossing property. This yields a simple…

We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix…

We study positive $m$-divisible non-crossing partitions and their positive Kreweras maps. In classical types, we describe their combinatorial realisations as certain non-crossing set partitions. We also realise these positive Kreweras maps…

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…

We consider noncrossing partitions of [n] under the action of (i) the reflection group (of order 2), (ii) the rotation group (cyclic of order n) and (iii) the rotation/reflection group (dihedral of order 2n). First, we exhibit a bijection…

We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular we study the restriction of these orders to noncrossing partitions and show…

In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of…

A set partition is said to be $(k,d)$-noncrossing if it avoids the pattern $12... k12... d$. We find an explicit formula for the ordinary generating function of the number of $(k,d)$-noncrossing partitions of $\{1,2,...,n\}$ when $d=1,2$.

Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and…

Let $A$ be a hereditary algebra. We construct a fundamental domain for the cluster category of $A$ inside the category of modules over the duplicated algebra $\bar{A}$ of $A$. We then prove that there exists a bijection between the tilting…

In this note we introduce several instructive examples of bijections found between several different combinatorially defined sequences of sets. Each sequence has cardinalities given by the Catalan numbers. Our results answer some questions…

It is well known that the number of distinct non-crossing matchings of $n$ half-circles in the half-plane with endpoints on the x-axis equals the $n^{th}$ Catalan number $C_n$. This paper generalizes that notion of linear non-crossing…

We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical…

We construct a direct natural bijection between descending plane partitions without any special part and permutations. The directness is in the sense that the bijection avoids any reference to nonintersecting lattice paths. The advantage of…

In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we identify Panyushev's map with the Kreweras complement on the set of noncrossing…