A Sperner-Type Theorem for Set-Partition Systems

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 $n$.

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A Deza–Frankl Type Theorem for Set Partitions

The Electronic Journal of Combinatorics ◽

10.37236/4987 ◽

2015 ◽

Vol 22 (1) ◽

Author(s):

Cheng Yeaw Ku ◽

Kok Bin Wong

Keyword(s):

Pairwise Disjoint ◽

Type Theorem ◽

Set Partition ◽

Set Partitions ◽

The Family

A set partition of $[n]$ is a collection of pairwise disjoint nonempty subsets (called blocks) of $[n]$ whose union is $[n]$. Let $\mathcal{B}(n)$ denote the family of all set partitions of $[n]$. A family $\mathcal{A} \subseteq \mathcal{B}(n)$ is said to be $m$-intersecting if any two of its members have at least $m$ blocks in common. For any set partition $P \in \mathcal{B}(n)$, let $\tau(P) = \{x: \{x\} \in P\}$ denote the union of its singletons. Also, let $\mu(P) = [n] -\tau(P)$ denote the set of elements that do not appear as a singleton in $P$. Let \begin{align*} {\mathcal F}_{2t} & =\left\{P \in \mathcal{B}(n)\ : \ \vert \mu (P)\vert\leq t\right\};\\{\mathcal F}_{2t+1}(i_0) & =\left\{P \in \mathcal{B}(n)\ : \ \vert\mu (P)\cap ([n]\setminus \{i_0\})\vert\leq t\right\}.\end{align*} In this paper, we show that for $r\geq 3$, there exists a $n_0=n_0(r)$ depending on $r$ such that for all $n\geq n_0$, if $\mathcal{A} \subseteq\mathcal{B}(n)$ is $(n-r)$-intersecting, then \[ |\mathcal{A}| \leq \begin{cases} \vert {\mathcal F}_{2t} \vert, & \text{if $r=2t$};\\ \vert {\mathcal F}_{2t+1}(1) \vert, & \text{if $r=2t+1$}.\end{cases}\]Moreover, equality holds if and only if \[ \mathcal{A}= \begin{cases} {\mathcal F}_{2t}, & \text{if $r=2t$};\\ {\mathcal F}_{2t+1}(i_0), & \text{if $r=2t+1$},\end{cases}\]for some $i_0\in [n]$.

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Erdős-Ko-Rado theorems for uniform set-partition systems

The Electronic Journal of Combinatorics ◽

10.37236/1937 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 5

Author(s):

Karen Meagher ◽

Lucia Moura

Keyword(s):

Higher Order ◽

Set Partition ◽

Set Partitions

Two set partitions of an $n$-set are said to $t$-intersect if they have $t$ classes in common. 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 the Erdős-Ko-Rado theorem for systems of pairwise $t$-intersecting uniform $k$-partitions of an $n$-set. We prove that for $n$ large enough, any such system contains at most $${1\over(k-t)!} {n-tc \choose c} {n-(t+1)c \choose c} \cdots {n-(k-1)c \choose c}$$ partitions and this bound is only attained by a trivially $t$-intersecting system. We also prove that for $t=1$, the result is valid for all $n$. We conclude with some conjectures on this and other types of intersecting partition systems.

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

The Electronic Journal of Combinatorics ◽

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.

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Liouville type theorem for higher-order elliptic system with Navier boundary condition

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/s00030-014-0284-z ◽

2014 ◽

Vol 22 (2) ◽

pp. 311-324

Author(s):

Weiwei Zhao ◽

Jinge Yang ◽

Sining Zheng

Keyword(s):

Boundary Condition ◽

Elliptic System ◽

Type Theorem ◽

Higher Order ◽

Liouville Type ◽

Liouville Type Theorem ◽

Navier Boundary Condition ◽

Order Elliptic System

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Universality on higher order Hardy spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700037965 ◽

2005 ◽

Vol 71 (1) ◽

pp. 17-28

Author(s):

L. Bernal-González ◽

A. Bonilla ◽

M. C. Calderón-Moreno

Keyword(s):

Unit Disk ◽

Hardy Spaces ◽

Composition Operators ◽

Type Theorem ◽

Higher Order ◽

Interpolating Sequences ◽

Bounded Functions

We prove a Seidel-Walsh-type theorem about the universality of a sequence of derivation-composition operators generated by automorphisms of the unit disk in the setting of the higher order Hardy spaces. Moreover, some related positive or negative assertions involving interpolating sequences and sequences between two tangent circles are established for the class of bounded functions in the unit disk. Our statements improve earlier ones due to Herzog and to the first and third authors.

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Crossings and Nestings in Set Partitions of Classical Types

The Electronic Journal of Combinatorics ◽

10.37236/392 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 5

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.

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Actions and Identities on Set Partitions

The Electronic Journal of Combinatorics ◽

10.37236/1992 ◽

2012 ◽

Vol 19 (1) ◽

Cited By ~ 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.

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