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

We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and…

The counting of partitions according to their genus is revisited. The case of genus 0 -- non-crossing partitions -- is well known. Our approach relies on two pillars: first a functional equation between generating functions, originally…

We provide a systematic derivation of cluster alphabets of finite types. The construction is based on a geometric realization of the generalized worldsheets by gluing and folding a pair of polygons. The cross ratios of the worldsheet z…

Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of…

We point out that a sequence of natural numbers is the dimension sequence of a subproduct system if and only if it is the cardinality sequence of a word system (or factorial language). Determining such sequences is, therefore, reduced to a…

One of the most well known random fractals is the so-called Fractal percolation set. This is defined as follows: we divide the unique cube in $\mathbb{R}^d$ into $M^d$ congruent sub-cubes. For each of these cubes a certain retention…

We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…

Assume that $G$ is a finite group and let $a$ and $b$ be non-negative integers. We define an undirected graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and…

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 study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category…

Globular Clusters (GCs) have historically been subdivided in either two (disk/halo) or three (disk/inner-halo/outer-halo) groups based on their orbital, chemical and internal physical properties. The qualitative nature of this subdivision…

In this paper we show a a proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by \(k\)…

Given a Stirling permutation w, we introduce the mesa set of w as the natural generalization of the pinnacle set of a permutation. Our main results characterize admissible mesa sets and give closed enumerative formulas in terms of rational…

Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help.…

The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well…

Let $Q$ be an acyclic quiver and $w \geq 1$ be an integer. Let $\mathsf{C}_{-w} (\mathbf{k} Q)$ be the $(-w)$-cluster category of $\mathbf{k} Q$. We show that there is a bijection between simple-minded collections in $\mathsf{D}^b…

Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard…

Biclustering is a problem in machine learning and data mining that seeks to group together rows and columns of a dataset according to certain criteria. In this work, we highlight the natural relation that quantum computing models like boson…

We extend so-called slit-slide-sew bijections to constellations and quasiconstellations. We present an involution on the set of hypermaps given with an orientation, one distinguished corner, and one distinguished edge leading away from the…

This paper addresses the problem of finding $Q_{m,t}\left(n\right)$, the number of possible ways to partition any member $n$ of the cyclic group $\mathbb{Z}/m\mathbb{Z}$ into $t$ distinct parts. When $m$ is odd, it was previously known that…