Related papers: M\"obius numbers of some modified generalized nonc…
We define and study noncommutative crossing partitions which are a generalization of non-crossing partitions. By introducing a new cover relation on binary trees, we show that the partially ordered set of noncommutative crossing partitions…
We study filters in the partition lattice formed by restricting to partitions by type. The M\"obius function is determined in terms of the easier-to-compute descent set statistics on permutations and the M\"obius function of filters in the…
We present the M\"{o}bius functions of several posets of annular noncrossing objects, namely a self-dual extension of the annular noncrossing permutations, minimal length annular partitioned permutations, and annular noncrossing partitons.
Reflection groups, geometry of the discriminant and noncrossing partitions. When W is a well-generated complex reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of…
We derive a formula for the expected number of blocks of a given size from a non-crossing partition chosen uniformly at random. Moreover, we refine this result subject to the restriction of having a number of blocks given. Furthermore, we…
In this article we give a simple, almost uniform proof that the lattice of noncrossing partitions associated with a well-generated complex reflection group is lexicographically shellable. So far a uniform proof is available only for Coxeter…
We study some combinatorial statistics defined on the set $NC^{(mton)}(n)$ of monotonically ordered non-crossing partitions of {1,...,n}, and on the set $NC_2^{(mton)}(2n)$ of monotonically ordered non-crossing pair-partitions of…
Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the…
In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…
We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between noncrossing partitions and clusters in this context, and show…
We prove that the generalised non-crossing partitions associated to well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, respectively by Bessis and Reiner.…
We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…
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…
Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative…
Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also,…
Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex…
The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the M\"obius function is given for the general abelian group, while in many special cases, including the case of…
We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…
We give combinatorial proofs of the formulas for the number of multichains in the $k$-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and M{\"u}ller. We also…
We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new…