scholarly journals On colored set partitions of type B n

2014 ◽  
Vol 12 (9) ◽  
Author(s):  
David Wang
Keyword(s):  
Generating Function ◽  
Asymptotic Expression ◽  
Type B ◽  
Set Partitions ◽  
Asymptotic Normal

AbstractGeneralizing Reiner’s notion of set partitions of type B n, we define colored B n-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored B n-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B n-partition. We find an asymptotic expression of the total number of colored B n-partitions up to an error of O(n −1/2log7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored B n-partitions.

scholarly journals Derangement Polynomials and Excedances of Type $B$

10.37236/81 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Robert L. Tang ◽  
Alina F. Y. Zhao
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Type B ◽  
Eulerian Polynomials ◽  
Basic Properties ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal

Based on the notion of excedances of type $B$ introduced by Brenti, we give a type $B$ analogue of the derangement polynomials. The connection between the derangement polynomials and Eulerian polynomials naturally extends to the type $B$ case. Using this relation, we derive some basic properties of the derangement polynomials of type $B$, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.

scholarly journals Crossings and Nestings in Set Partitions of Classical Types

10.37236/392 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Martin Rubey ◽  
Christian Stump
Keyword(s):  
Type A ◽  
The Other ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Type D ◽  
Other Hand ◽  
The One

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.

scholarly journals Actions and Identities on Set Partitions

10.37236/1992 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Eric Marberg
Keyword(s):  
Abelian Group ◽  
Group Action ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Unitriangular Group

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $\mathbb{A}$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $\mathbb{A}^n$ on the set of $\mathbb{A}$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D.

scholarly journals Counting water cells in bargraphs of compositions and set partitions

2018 ◽  
Vol 12 (2) ◽  
pp. 413-438 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Horizontal Line ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Unit Square ◽  
Joint Distributions ◽  
Water Cell ◽  
First Moment

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

scholarly journals Minimally Intersecting Set Partitions of Type $B$

10.37236/294 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
William Y.C. Chen ◽  
David G.L. Wang
Keyword(s):  
Type B ◽  
Set Partitions

Motivated by Pittel's study of minimally intersecting set partitions, we investigate minimally intersecting set partitions of type $B$. Our main result is a formula for the number of minimally intersecting $r$-tuples of $B_n$-partitions. As a consequence, it implies the formula of Benoumhani for the Dowling number in analogy to Dobiński's formula.

scholarly journals The Perimeter and Site-Perimeter of Set Partitions

10.37236/8250 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Explicit Formulas ◽  
Set Partitions

In this paper, we study the generating function for the number of set partitions of $[n]$ represented as bargraphs according to the perimeter/site-perimeter. In particular, we find explicit formulas  for the total perimeter and the total site-perimeter over all set partitions of $[n]$.

scholarly journals Excedance Numbers for the Permutations of Type B

10.37236/2375 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Alina F.Y. Zhao
Keyword(s):  
Fixed Points ◽  
Generating Function ◽  
Type B ◽  
Euler Numbers ◽  
Combinatorial Construction

This work provides a study on the multidistribution of type $B$ excedances, fixed points and cycles on the permutations of type $B$. We derive the recurrences and closed formulas for the distribution of signed excedances on type $B$ permutations as well as derangements via combinatorial construction. Based on this result, we obtain the recurrence and generating function for the signed excedance polynomial and disclose some relationships with Euler numbers and Springer numbers, respectively.

scholarly journals Two by two squares in set partitions

2020 ◽  
Vol 70 (1) ◽  
pp. 29-40
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Set Partitions ◽  
Minimal Elements ◽  
Growth Functions ◽  
Asymptotic Formulae ◽  
Canonical Order ◽  
First Occurrence ◽  
Canonical Representations ◽  
The Mean

AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

scholarly journals Some Remarks on the Joint Distribution of Descents and Inverse Descents

10.37236/2135 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Mirkó Visontai
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Nonnegative Integer ◽  
Joint Distribution ◽  
Type B ◽  
Integer Coefficients ◽  
Set Of Permutations ◽  
Combinatorial Model

We study the joint distribution of descents and inverse descents over the set of permutations of $n$ letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture.  We examine generalizations of the conjecture and obtain a type $B$ analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.

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