Igor Pak, Greta Panova
We present three different upper bounds for Kronecker coefficients $g(λ,μ,ν)$ in terms of Kostka numbers, contingency tables and Littlewood--Richardson coefficients. We then give various examples, asymptotic applications, and compare them with existing lower bounds.
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We present several upper and lower bounds on the Kronecker coefficients of the symmetric group. We prove $k$-stability of the Kronecker coefficients generalizing the (usual) stability, and giving a new upper bound. We prove a lower bound via the characters of $S_n$. We apply these and other results to generalize Sylvester's unimodality of the $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in $q$: we derive explicit sharp bounds on the differences of their consecutive coefficients.
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…
A combinatorial interpretation for certain plethysm and Kronecker coefficients
本文给出了当 λ 至多两行时 plethysm 系数 ⟨s_μ[s_ν], s_λ⟩ 的显式正组合解释，通过计数特定标记树；特例 μ=(n) 也给出了矩形 Kronecker 系数 g(λ, (n^k), (n^k)) 的组合解释。
Vector partition functions and Kronecker coefficients
本文利用多面体几何工具将Kronecker函数表示为向量分拆函数的系数，定义原子Kronecker系数，并在首个非平凡情形给出精确公式与上界。
The Boolean rank of a $0,1$-matrix $A$, denoted $R_\mathbb{B}(A)$, is the smallest number of monochromatic combinatorial rectangles needed to cover the $1$-entries of $A$. In 1988, de Caen, Gregory, and Pullman asked if the Boolean rank of the Kronecker product $C_n \otimes C_n$ is strictly smaller than the square of $R_\mathbb{B}(C_n)$, where $C_n$ is the $n \times n$ matrix with zeros on the diagonal and ones everywhere else (Carib. Conf. Comb. & Comp., 1988). A positive answer was given by Watts for $n=4$ (Linear Alg. and its Appl., 2001). A result of Karchmer, Kushilevitz, and Nisan, motiv…
We prove that the semi-invariant ring of the standard representation space of the $l$-flagged $m$-arrow Kronecker quiver is an upper cluster algebra for any $l,m\in \mathbb{N}$. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient $g_{μ,ν}^λ$ with $λ$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the ${\rm g}$-vector cone, which is explicitly given.
In this paper we give a combinatorial interpretation for the coefficient of $s_ν$ in the Kronecker product $s_{(n-p,p)}\ast s_λ$, where $λ=(λ_1, ..., λ_{\ell(λ)})\vdash n$, if $\ell(λ)\geq 2p-1$ or $λ_1\geq 2p-1$; that is, if $λ$ is not a partition inside the $2(p-1)\times 2(p-1)$ square. For $λ$ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all $λ$ when $n>(2p-2)^2$. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker produ…
Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem
本文首次给出Kronecker系数的显式公式，揭示了参数值5处初等结构崩溃的普遍边界，通过整数强制技术证明了首个三行Kronecker系数的显式公式，并验证了Saxl猜想。
Kronecker Products, Polarity Quotients and Large Graph Constructions
本文建立带极性二部图克罗内克积与极性商的结构兼容性，应用该理论以广义多边形为因子构造出三类大图，其中直径3的低度数图大小超现有同类图。
The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that need to be changed in order to obtain a matrix of rank at most $r$. At MFCS'77, Valiant introduced matrix rigidity as a tool to prove circuit lower bounds for linear functions and since then this notion received much attention and found applications in other areas of complexity theory. The problem of constructing an explicit family of matrices that are sufficiently rigid for Valiant's reduction (Valiant-rigid) still remains open. Moreover, since 2017 most of the long-studied candidates have been shown…
For a matrix $M$ and a positive integer $r$, the rank $r$ rigidity of $M$ is the smallest number of entries of $M$ which one must change to make its rank at most $r$. There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include: $\bullet$ For any $d> 1$, and over any field $\mathbb{F}$, the $N \times N$ Walsh-Hadamard transform has a depth-$d$ linear circuit of size $O…
The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = ω(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the maximum number of ones and the maximum number of columns in a minimally non-linear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the bound on the maximum number of ones in a minimally non-linear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary…
We study asymptotics of an irreducible representation of the symmetric group S_n corresponding to a balanced Young diagram λ(a Young diagram with at most C\sqrt{n} rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that |χ^λ(π)| = | Tr ρ^λ(π)/Tr ρ^λ(e) | < [ D max(1,|π|^2/n) / \sqrt{n}} ]^{|π|}, where |π| denotes the length of a permutation (the minimal number of factors necessary to write πas a product of transpositions). Our main tool is an analogue of Frobenius character formula wh…
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic results, and it depends only on the number of rows of the coefficient matrix and the row densities. We also consider the particular case of matrices that are \textit{almost} block decomposable, and show how the bound may improve according to the particular decomposition adopted. Such final result may provide interesting indications for comparing different matrix de…
We study the maximal number of pairwise distinct columns in a $Δ$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $Δ$. We complement this and obtain an upper bound of the form $O(Δ)$ for fixed $m$, and with the implied constant depending polynomially on $m$.
Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask if there is a minimum integer $\ell=\ell(n,k)$ such that for all $m>\ell$, any $m\times n$ level matrix with entries in $\{0,\ldots,k\}$ is reducible. It is known that $\ell(2,k)=2k-1$. In this paper, we establish the existence of $\ell(n,k)$ for $n\geq 3$ by giving upper and lower bounds for it. We then apply this result to bound the number of certain types of vec…
A covering array $\mathsf{CA}(N;t,k,v)$ is an $N\times k$ array with entries in $\{1, 2, \ldots , v\}$, for which every $N\times t$ subarray contains each $t$-tuple of $\{1, 2, \ldots , v\}^t$ among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound $\mathsf{CAN}(t,k,v)$, the minimum number $N$ of rows of a $\mathsf{CA}(N;t,k,v)$. The well known bound $\mathsf{CAN}(t,k,v)=O((t-1)v^t\log k)$ is not too far from being asymptotically optimal.…
A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in l_p is attained either for the identity matrix I or for a constant multiple of the all-1 matrix J. The conjecture is known to be true for p = 1 (I) and for p \ge 2 (J). We prove the conjecture for a subinterval of (1,2), and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1 < p < 2. In fact, for p bounded away from 1, the conjectured upper bound is true within a constan…
Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, CAN$(t,k,v)$, in a covering array for given values of the parameters $t,k$ and $v$. Asymptotic upper bounds for CAN$(t,k,v)$ have earlier been established using the Stein-Lovász-Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lovász-Johnson bound is derived. Then using alteration, the Stein-Lovász-Johnson bound i…
On three conjectures of Kimberling concerning the array $\lfloor kφ^n\rfloor$
本文证明了 Kimberling 关于数组 $\lfloor k\varphi^n\rfloor$ 的三个猜想：前两个由 Skolem-Bang 定理推出，第三个通过确定包含奇下标 Lucas 数的行及 Lucas 移位引理证明，并给出计算证据。
Covering arrays are generalizations of orthogonal arrays that have been widely studied and are used in software testing. The probabilistic method has been employed to derive upper bounds on the sizes of minimum covering arrays and give asymptotic upper bounds that are logarithmic on the number of columns of the array. This corresponds to test suites with a desired level of coverage of the parameter space where we guarantee the number of test cases is logarithmic on the number of parameters of the system. In this paper, we study variable strength covering arrays, a generalization of covering ar…
We prove an upper bound for characters of the symmetric groups. Namely, we show that there exists a constant a>0 with a property that for every Young diagram λwith n boxes, r(λ) rows and c(λ) columns |Tr ρ^λ(π) / Tr ρ^λ(e)| < [a max(r(λ)/n, c(λ)/n,|π|/n) ]^{|π|}, where |π| is the minimal number of factors needed to write π\in S_n as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov's and Biane's character formulas and give a new formula for free cumulants of the transition measure.
Upper Bounds for Covering Arrays of Higher Index
本文研究一般指数 λ 的覆盖阵列数上界，结合 Stein–Lovász–Johnson 范式、Lovász 局部引理、两阶段构造与图着色，消去 log log k 项并改进常数及 λ=2、一般 λ 的界。
Bounds for Characters of the Symmetric Group: A Hypercontractive Approach
本文通过超收缩性方法改进了对称群特征比的上界，利用规范理论工具扩展了方法的应用范围，无需依赖特定于S_n的Murnaghan-Nakayama规则。
A binary matrix is a matrix with entries from the set $\{0,1\}$. We say that a binary matrix $A$ contains a binary matrix $S$ if $S$ can be obtained from $A$ by removal of some rows, some columns, and changing some $1$-entries to $0$-entries. If $A$ does not contain $S$, we say that $A$ avoids $S$. A $k$-permutation matrix $P$ is a binary $k \times k$ matrix with exactly one $1$-entry in every row and one $1$-entry in every column. The Füredi-Hajnal conjecture, proved by Marcus and Tardos, states that for every permutation matrix $P$, there is a constant $c_P$ such that for every $n \in \mathb…
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