Related papers: A simple bijection for enhanced, classical, and 2-…

A bijection is presented between (1): partitions with conditions $f_j+f_{j+1}\leq k-1$ and $ f_1\leq i-1$, where $f_j$ is the frequency of the part $j$ in the partition, and (2): sets of $k-1$ ordered partitions $(n^{(1)}, n^{(2)}, ...,…

Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural…

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…

This note discusses the bijection between the exceptional subcategories of representations of quivers and generalized non-crossing partitions of Weyl groups. We give a new proof of the Ingalls-Thomas-Igusa-Schiffler bijection by using the…

We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.

We generalize overpartitions to (k,j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems:…

We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two…

For each finite configuration of distinct points in the plane, there is an associated lattice of noncrossing partitions. When these points form the vertices of a convex polygon, the result is the classical noncrossing partition lattice,…

We give three proofs of the following result conjectured by Carriegos, De Castro-Garc\'{\i}a and Mu\~noz Casta\~neda in their work on enumeration of control systems: when $\binom{k+1}{2} \le n < \binom{k+2}{2}$, there are as many partitions…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

We show that the poset of non-trivial partitions of 1,2,...,n has a fundamental homology class with coefficients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations…

We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower…

It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain…

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…

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…

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank <= k and those with k in…

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…

The study of flattened partitions is an active area of current research. In this paper, our study unexpectedly leads us to the OEIS numbers A124324. We provide a new combinatorial interpretation of these numbers. A combinatorial bijection…

We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification…

Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate this number from above by $|X|^{c(n)}$ where $$ c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n}…