Given a $d$-dimensional tensor $T:A_1\times\dots\times A_d\rightarrow \mathbb{F}$ (where $\mathbb{F}$ is a field), the $i$-flattening rank of $T$ is the rank of the matrix whose rows are indexed by $A_{i}$, columns are indexed by $B_{i}=A_1\times\dots\times A_{i-1}\times A_{i+1}\times\dots\times A_{d}$ and whose entries are given by the corresponding values of $T$. The max-flattening rank of $T$ is defined as $\text{mfrank}(T)=\max_{i\in [d]}\text{frank}_{i}(T)$. A tensor $T:A^{d}\rightarrow\mathbb{F}$ is called semi-diagonal, if $T(a,\dots,a)\neq 0$ for every $a\in A$, and $T(a_{1},\dots,a_{d})=0$ for every $a_{1},\dots,a_{d}\in A$ that are all distinct. In this paper we prove that if $T:A^{d}\rightarrow\mathbb{F}$ is semi-diagonal, then $\text{mfrank}(T)\geq \frac{|A|}{d-1}$, and this bound is the best possible.
We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an $r$-uniform multi-hypergraph $\mathcal{H}$ are colored with $z$ colors such that each colorclass is a matching of size $t$, then $\mathcal{H}$ contains a rainbow matching of size $t$ provided $z>(t-1)\binom{rt}{r}$. This improves previous results of Alon and Glebov, Sudakov and Szabó.
The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix $B \in \mathbb{F}^{n \times n}$ is defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_j \in \{\tt{A}, \tt{S}, \tt{N}\}$ according to whether all, some but not all, or none of the principal minors of order $j$ of $B$ are nonzero. Building upon the second author's recent classification of the epr-sequences of symmetric matrices over the field $\mathbb{F}=\mathbb{F}_2$, we initiate a study of the case $\mathbb{F}=\mathbb{F}_3$. Moreover, epr-sequences over finite fields are shown to have connections to Ram…
The partition statistic $V_R$-rank is introduced to give combinatorial proofs of the Ramanujan type congruences mod 3 for certain classes of partition functions.
In this paper, we calculate the dimension of root spaces $\mathfrak{g}_λ$ of a special type rank $3$ Kac-Moody algebras $\mathfrak{g}$. We first introduce a special type of elements in $\mathfrak{g}$, which we call elements in standard form. Then, we prove that any root space is spanned by these elements. By calculating the number of linearly independent elements in standard form, we obtain a formula for the dimension of root spaces $\mathfrak{g}_λ$, which depends on the root $λ$.
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{CT}_{n}$ be the semigroup of full contractions on $[n]$. Denote $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$ to be the subsemigroup of order preserving or reversing and the subsemigroup of order preserving full contractions, respectively. It was shown in [17] that the collection of all regular elements (denoted by, Reg$(\mathcal{ORCT}_{n})$ and Reg$(\mathcal{OCT}_{n}$), respectively) and the collection of all idempotent elements (denoted by E$(\mathcal{ORCT}_{n})$ and E$(\mathcal{OCT}_{n}$), respectively) of the subsemigroups $\mat…
We give a combinatorial proof of a recent result of Bóna by constructing a bijection from the set of all neighbors of leaves of increasing trees of size $n$ to the set of derangements of length $n$.
In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{αβ} x_{αβ})$, where $A_{αβ}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{αβ}$ is an indeterminate for $α= 1,2,\dots, μ$ and $β= 1,2, \dots, ν$. This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018) and Garg, Gurvits, Oliveiva…
We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we sh…
Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required…
Matroid is a generalization of many fundamental objects in combinatorial mathematics , and matroid intersection problem is a classical subject in combinatorial optimization . However , only the intersection of two matroids are well understood . The solution of the intersection problem of more than three matroids is proved to be \textbf{NP-hard} . We will give a lower bound estimate on the maximal cardinality of the common independent sets in matroid intersections . We will also study some properties of the intersection of more than two matroids and deduce some analogous results for Edmonds' Mi…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.
We study combinatorial and asymptotic properties of the rank of strongly unimodal sequences. We find a generating function for the rank enumeration function, and give a new combinatorial interpretation of the ospt-function introduced by Andrews, Chan, and Kim. We conjecture that the enumeration function for the number of unimodal sequences of a fixed size and varying rank is log-concave, and prove an asymptotic result in support of this conjecture. Finally, we determine the asymptotic behavior of the rank for strongly unimodal sequences, and prove that its values (when appropriately renormaliz…
We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new …
The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for finite fields $\mathbb{F}_q$ with $q$ odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson's classification of nets of rank one, namely those cont…
