scholarly journals Average Values of some Z-Parameters in a Random Set Partition

10.37236/715 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Anisse Kasraoui
Keyword(s):  
Asymptotic Formulas ◽  
Random Set ◽  
Set Partition ◽  
Set Partitions

We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.

scholarly journals Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

10.37236/2550 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Adam M Goyt ◽  
Brady L Keller ◽  
Jonathan E Rue
Keyword(s):  
Fibonacci Numbers ◽  
Natural Extension ◽  
Statistical Distributions ◽  
Set Partition ◽  
Set Partitions ◽  
Partition Statistics

We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k.  The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics.  We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers.  This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations.  In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles.  We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

scholarly journals On the maximal multiplicity of block sizes in a random set partition

2020 ◽  
Vol 56 (3) ◽  
pp. 867-891
Author(s):  
Ljuben R. Mutafchiev ◽  
Mladen Savov
Keyword(s):  
Random Set ◽  
Set Partition ◽  
Maximal Multiplicity ◽  
Block Sizes

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 A Sperner-Type Theorem for Set-Partition Systems

10.37236/1987 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Karen Meagher ◽  
Lucia Moura ◽  
Brett Stevens
Keyword(s):  
Type Theorem ◽  
Higher Order ◽  
Set Partition ◽  
Set Partitions ◽  
Uniform Partition ◽  
Partition System ◽  
Sperner's Theorem

A Sperner partition system is a system of set partitions such that any two set partitions $P$ and $Q$ in the system have the property that for all classes $A$ of $P$ and all classes $B$ of $Q$, $A \not\subseteq B$ and $B \not\subseteq A$. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of Sperner's Theorem. In particular, we show that if $k$ divides $n$ the largest Sperner $k$-partition system on an $n$-set has cardinality ${n-1 \choose n/k-1}$ and is a uniform partition system. We give a bound on the cardinality of a Sperner $k$-partition system of an $n$-set for any $k$ and $n$.

scholarly journals Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

10.37236/8241 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Megan Ly
Keyword(s):  
Irreducible Representations ◽  
Character Theory ◽  
Set Partition ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Weyl Duality ◽  
Supercharacter Theory ◽  
Combinatorial Representation Theory ◽  
Centralizer Algebra

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field.  In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph. 

Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions

Integers ◽  
2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Mark Shattuck
Keyword(s):  
Set Partition ◽  
Set Partitions ◽  
Finite Set

AbstractA finite set partition is said to have a

scholarly journals Extensions of Set Partitions and Permutations

10.37236/8223 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Jhon B. Caicedo ◽  
Victor H. Moll ◽  
José L. Ramírez ◽  
Diego Villamizar
Keyword(s):  
Set Partition ◽  
Set Partitions ◽  
Combinatorial Properties

Extensions of a set partition obtained by imposing bounds on the size of the parts is examined. Arithmetical and combinatorial properties of these sequences are established.

Random Set Partitions

1994 ◽  
Vol 7 (3) ◽  
pp. 419-436 ◽  
Author(s):  
William M. Y. Goh ◽  
Eric Schmutz
Keyword(s):  
Random Set ◽  
Set Partitions
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