Related papers: Some Families of Type $B$ Set Partitions Counted b…
We show that the joint distribution of the number of singleton pairs and the number of adjacency pairs is symmetric over the set partitions of type $B_n$ without zero-block, in analogy with the result of Callan for ordinary partitions.
Stirling numbers, which count partitions of a set and permutations in the symmetric group, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the…
Motivated by Pittel's study of minimally intersecting set partitions, we investigate minimally intersecting set partitions of type B. We find a formula for the number of minimally intersecting r-tuples of $B_n$-partitions, as well as a…
In this paper, we study merging-free partitions with their canonical forms and run-sorted permutations. We give a combinatorial proof of the conjecture made by Nabawanda et al. We describe the distribution of the statistics of runs and…
It is a classical result of Harper that the limiting distribution of the number of blocks in partitions of the set $\{1, 2,..., n\}$ is normal. In this paper, using the saddle point method we prove the normality of the limiting distribution…
Descending plane partitions, alternating sign matrices, and totally symmetric self-complementary plane partitions are equinumerous combinatorial sets for which no explicit bijection is known. In this paper, we isolate a subset of descending…
We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $\Gamma_1,\ldots,\Gamma_j$ of low-degree…
We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the q-Stirling numbers of the second kind. Some of…
A $k$-Stirling permutation of order $n$ is said to be "flattened" if the leading terms of its increasing runs are in ascending order. We show that flattened $k$-Stirling permutations of order $n+1$ are in bijection correspondence with a…
Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…
In this paper, we study gamma-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present grammatical proofs of the gamma-positivity of the Eulerian polynomials, type B Eulerian…
The distribution of descents in a fixed conjugacy class of $S_n$ is studied, and it is shown that its moments have an interesting property. A particular conjugacy class that is of interest is the class of matchings (also known as fixed…
We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study…
A \emph{set partition} of the set $[n]=\{1,...c,n\}$ is a collection of disjoint blocks $B_1,B_2,...c, B_d$ whose union is $[n]$. We choose the ordering of the blocks so that they satisfy $\min B_1<\min B_2<...b<\min B_d$. We represent such…
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…
To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each…
Recall that a Stirling permutation is a permutation on the multiset $\{1,1,2,2,\ldots,n,n\}$ such that any numbers appearing between repeated values of $i$ must be greater than $i$. We call a Stirling permutation ``flattened'' if the…
We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter…
Combinatorial identities on Weyl groups of types $A$ and $B$ are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type $A$ we give a new construction…
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…