Related papers: A bijection between noncrossing and nonnesting par…
Our aim is to describe the theory of Cartesian decompositions preserved by some member of a large family of finite transitive permutation groups called innatelytransitive groups.
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…
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,…
A k-triangulation of a convex polygon is a maximal set of diagonals so that no k+1 of them mutually cross in their interiors. We present a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length…
There is a received wisdom about where to draw the boundary between classical and nonclassical for various types of quantum processes. For multipartite states, it is the divide between separable and entangled; for channels, the divide…
We present a bijection between cyclic permutations of {1,2,...,n+1} and permutations of {1,2,...,n} that preserves the descent set of the first n entries and the set of weak excedances. This non-trivial bijection involves a Foata-like…
We show that any finite group $G$ there exists a bijction $f$ from $G$ onto $C_{n}$ such that $o(x)$ divides $o(f(x))$ for all $x\in G$. This confirm Problem 18.1 in [7].
We give a bijection between ordered $m$-clusters and (complete) $m$-exceptional sequences, a concept that we introduce for this purpose. This holds for all hereditary artin algebras. This extends the bijection in the $m = 1$ case shown in…
Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in…
We examine an elementary problem on prime divisibility of binomial coefficients. Our problem is motivated by several related questions on alternating groups.
We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called $\otimes ^c$-weight structures.…
We study the structure of two cointeracting bialgebras on noncrossing partitions appearing in the theory of free probability. The first coproduct is given by separation of the blocks of the partitions into two parts, with respect to the…
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the…
Let $A$ be a hereditary algebra. We construct a fundamental domain for the cluster category of $A$ inside the category of modules over the duplicated algebra $\bar{A}$ of $A$. We then prove that there exists a bijection between the tilting…
The partition function for a system of non-interacting $N-$particles can be found by summing over all the states of the system. The classical partition function for an ideal gas differs from Bosonic or Fermionic partition function in the…
The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum_{u,w\in P} ^{}\mu(u,w) x^{\rk u}y^{\rk w}$, where $\mu(.,.)$ is the M\"obius function of $P$. We compute the $M$-triangle of Armstrong's poset of…
We introduce the notion of a $\textit{reflection fusion category}$, which is a type of a $G$-crossed category generated by objects of Frobenius-Perron dimension $1$ and $\sqrt{p}$, where $p$ is an odd prime. We show that such categories…
Categories of partial functions have become increasingly important principally because of their applications in theoretical computer science. In this note we prove that the category of partial bijections between sets as an…
For affine Coxeter groups of affine types $\tilde D$ and $\tilde B$, we model the interval $[1,c]_T$ in the absolute order by symmetric noncrossing partitions of an annulus with one or two double points. In type $\tilde B$ (and…
For modules over a finite-dimensional algebra, there is a canonical one-to-one correspondence between the projective indecomposable modules and the simple modules. In this purely expository note, we take a straight-line path from the…