A \textsl{Sperner $k$-partition system} on a set $X$ is a set of partitions of $X$ into $k$ classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another partition). These systems were defined by Meagher, Moura and Stevens in \cite{MMS} who showed that if $|X| = k \ell$, then the largest Sperner $k$-partition system has size $\frac{1}{k}\binom{|X|}{\ell}$. In this paper we find bounds on the size of the largest Sperner $k$-partition system where $k$ does not divide the size of $X$, specifically, we give an exact bound when $k=2$ and upper and lower bounds when $|X| = 2k+1$, $|X|=2k+2$ and $|X| = 3k-1$.
An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems based on a division of the ground set into many equal-sized parts. We use this to asymptotically determine $\mathrm{SP}(n,k)$ in many cases where $\frac{n}{k}$ is bounded as $n$ becomes large. Further, we show that this c…
Covering arrays on graphs: qualitative independence graphs and extremal set partition theory
本文主要研究覆盖数组的推广，覆盖数组在图上的推广。Z_k^n中的两个向量v，w是定性独立的，如果对Z_k x Z_k中的所有序偶（a，b），向量中存在位置i，其中（a，b）=（v_i，w_i）。覆盖数组是一个具有任何一对行是定性独立的性质的数组。图的覆盖数组是指图的每个顶点都有一行的数组，且具有这样的性质：对应于邻顶点的任意两行是定性独立的。在覆盖数组中添加图结构使得使用图论的方法来研究这些设计成为可能。在这篇论文中，我们定义了一类图，称为定性独立图。一个图有给定参数的覆盖阵列，当且仅当存在从该图到一个特定的定性独立图的同态。定性独立图中的团与覆盖数组有关，独立集与相交划分系统有关。已知最优二元覆盖阵列的确切大小可以使用Sperner定理和Erdos-Ko-Rado定理来确定。由于一般覆盖阵列的行对应于集合划分，我们给出了Sperner定理和Erdos-Ko-Rado定理在集合划分系统中的推广。我们还考虑了一般定性独立图的一个子图，称为一致定性独立图。我们给出了这些图中的几个图的谱，并猜想它们是一个关联图。我们还给出了一个新的覆盖阵列的结构，它产生了许多新的最佳覆盖阵列的大小的上界。
A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erdős-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3…
In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee rectangle. We also obtain a new bijective proof of a result of Andrews and Bressoud that the number of partitions of $N$ with all ranks at least $1-\ell$ equals the number of partitions of $N$ with no parts equal to $\ell+1$, for $\ell\ge0$, which allows us to refine it by the above statistics. Combining Foata's second fundamental transformation for words with Greene…
For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges having ends in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-edge-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of graphs with no $K_4\setminus e$ topological minors in terms of forbidden obstructions. We also define $\mathcal C$-vertex-brittleness of a graph $G$ as the minimum $\ell$ such that the edge …
In this paper we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type $C^{(1)}_\ell$, $\ell\geq2$, and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.
For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case of a conjecture of Keith and Zanello on self-similarities of $b_3(n)$ modulo $2$. We next prove that the series $\sum_{n=0}^{\infty}b_9(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$. We also prove that the series $\sum_{n=0}^{\infty}b_9(4n)q^n$ is lacunary modulo $2$.
The partition relation N \to (n)_{ \ell }^k means that whenever the k-tuples of an N-element set are \ell -colored, there is a monochromatic set of size n, where a set is called monochromatic if all its k-tuples have the same color. The logical negation of N \to (n)_{ \ell }^k is written as N \not \to (n)_{ \ell }^k. An ingenious construction of Erdős and Hajnal known as the stepping-up lemma gives a negative partition relation for higher uniformity from one of lower uniformity, effectively gaining an exponential in each application. Namely, if \ell \geq 2, k \geq 3, and N \not \to (n)_{ \ell …
Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results between the number of partitions of an integer $n$ whose parts are taken from the sequence $\{f(\ell)\}_{\ell=1}^{\infty}$ and the number of parts of those partitions which are in a certain arithmetic progression.
Partitioning set $[n] = \{1, \dots, n\}$ into subsets of size at most $m$ such that all sums are powers of $m$
National Research University Higher School of Economics (HSE), Moscow, Russia（国立研究型高等经济大学）；RUTCOR, Rutgers University, Piscataway, NJ, United States（罗格斯大学运筹中心）；Business School, Rutgers University, Piscataway, NJ, United States（罗格斯大学商学院）
Let $κ$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be given. Let $p_{κ, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(\lfloor \ell^κ\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{κ, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution.
We study the Kronecker coefficients $g_{λ, μ, ν}$ via a formula that was described by Mishna, Rosas, and Sundaram, in which the coefficients are expressed as a signed sum of vector partition function evaluations. In particular, we use this formula to determine formulas to evaluate, bound, and estimate $g_{λ, μ, ν}$ in terms of the lengths of the partitions $λ, μ$, and $ν$. We describe a computational tool to compute Kronecker coefficients $g_{λ, μ, ν}$ with $\ell(μ) \leq 2,\ \ell(ν) \leq 4,\ \ell(λ) \leq 8$. We present a set of new vanishing conditions for the Kronecker coefficients by relatin…
We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|Π_n|$, which is the $Σ_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|Π_n|^G$ in terms of subgroup posets for general $G\subset Σ_n$ and prove a formula for the restriction of $|Π_n|$ to Young subgroups $Σ_{n_1}\times \dots\times Σ_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover su…
We let $A=R/I$ be a standard graded Artinian algebra quotient of $R={\sf k}[x,y]$, the polynomial ring in two variables over a field ${\sf k}$ by an ideal $I$, and let $n$ be its vector space dimension. The Jordan type $P_\ell$ of a linear form $\ell\in A_1$ is the partition of $n$ determining the Jordan block decomposition of the multiplication on $A$ by $\ell$ -- which is nilpotent. The first three authors previously determined which partitions of $n=\dim_{\sf k}A$ may occur as the Jordan type for some linear form $\ell$ on a graded complete intersection Artinian quotient $A=R/(f,g)$ of $R$,…
