Related papers: Noncommutative crossing partitions

In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee…

The notion of exponential Dowling structures is introduced, generalizing Stanley's original theory of exponential structures. Enumerative theory is developed to determine the M\"obius function of exponential Dowling structures, including a…

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.…

A natural first step in the classification of all `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of…

In this article we introduce the study of the number of pairs of non-comparable elements in a distributive lattice $\L$. We give several tight lower and upper bounds for the number and give as an application the lattices precisely for which…

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…

When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how…

For a finite real reflection group $W$ we use non-crossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$-discriminant $\Delta_W$ and that of the Milnor fiber of…

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case)…

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of…

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all…

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting…

Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya's work) to classify exceptional sequences of representations of Q,…

We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of…

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type $D_n$ and those of exceptional type and rank…

In this article we tackle the combinatorics of coloured hard-dimer objects. This is achieved by identifying coloured hard-dimer configurations with a certain class of rooted trees that allow for an algebraic treatment in terms of…

We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing…

We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution…

We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block…

We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the…