Related papers: M\"obius numbers of some modified generalized nonc…
Consider the noncrossing set partitions of an $n$-element set which either do not contain the block $\{n-1,n\}$, or which do not contain the singleton block $\{n\}$ whenever $1$ and $n-1$ are in the same block. In this article we study the…
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…
We present an elementary type preserving bijection between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis.
Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $\alpha\in R$, $v,w\in W$, we denote by $\partial_\alpha$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew…
In important work on the parity of the partition function, Ono related values of the partition function to coefficients of a certain mock theta function modulo 2. In this paper, we use M\"obius inversion to give analogous results which…
In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the…
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…
We prove a conjecture of Drake and Kim: the number of $2$-distant noncrossing partitions of $\{1,2,...,n\}$ is equal to the sum of weights of Motzkin paths of length $n$, where the weight of a Motzkin path is a product of certain fractions…
We obtain a formula for the M\"obius number of the socle of any group. Specifically, the M\"obius number of a direct product of simple groups is computed in terms of the M\"obius numbers of the simple groups which make up the product.
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…
In the combinatorics of finite finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpretation of a result by Deligne which is due to Chapoton, and the goal of this…
In this paper, we present a generalization of one of the theorems in [G. E. Andrews, Partitions with parts separated by parity, \textit{Annals of Combinatorics} \textbf{23}(2019), 241 - 248], and give its bijective proof. Further variations…
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…
For a fixed integer $k$, we consider the set of noncrossing partitions, where both the block sizes and the difference between adjacent elements in a block is $1\bmod k$. We show that these $k$-indivisible noncrossing partitions can be…
We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the…
For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for…
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…
We determine the rank generating function, the zeta polynomial and the Moebius function for the poset NC^B(p,q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some…
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear…
By a re-examination of MacMahon's original proof of his celebrated theorem on the distribution of the major indices over permutations, we give a reformulation of his argument in terms of the structure of labeled partitions. In this…