Related papers: A bijection between noncrossing and nonnesting par…

We give a criterion for Bruhat order on noncrossing partitions corresponding to the Coxeter element $c=s_1 s_2\cdots s_n$. Using it we prove that the Bruhat order endows noncrossing partitions with a lattice structure. We then explain what…

This paper generalizes in the affine symmetric group the notion of Coxeter sortable (or c-sortable for short) elements, as well as the classical bijection between c-sortable elements and c-noncrossing partitions defined by Reading in finite…

Schutzenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an…

In combinatorics there is a well-known duality between non-nesting and non-crossing objects. In algebra there are many objects which are standard, for example Standard Young Tableaux, Standard Monomials, Standard Bitableaux. We adopt a…

We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d,…

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different…

We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in…

Let $\pi$ be a set partition of $[n]=\{1,2,...,n\}$. The standard representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block which does not contain any integer…

We give an asymptotic estimate for the number of partitions of a set of $n$ elements, whose block sizes avoid a given set $\mathcal{S}$ of natural numbers. As an application, we derive an estimate for the number of partitions of a set with…

The present thesis studies structural properties of non-crossing partitions associated to finite Coxeter groups from both algebraic and geometric perspectives. On the one hand, non-crossing partitions are lattices, and on the other hand, we…

For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a…

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…

We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of…

In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a…

A bijection $\Phi$ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the…

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…

We define noncrossing partitions of a marked surface without punctures (interior marked points). We show that the natural partial order on noncrossing partitions is a graded lattice and describe its rank function topologically. Lower…

It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain…

The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked partitions introduced by Dykema in 2007 to study the S-transform, and the partial order << on NC(n) introduced by…

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…