相关论文: Latin transversals of rectangular arrays

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called…

We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other…

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique…

Let A be an n by n matrix with entries in an arbitrary field, and c_1,...,c_n be scalars. We prove that if A is not a scalar multiple of the identity matrix, then the condition c_1+...+c_n=tr(A) is necessary and sufficient for A to be…

The Dinitz conjecture states that, for each $n$ and for every collection of $n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for $(n-1)\times…

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for…

For Latin squares the units (rows and columns) have fixed sum. The same holds for rows, columns, and blocks in Sudokus. Summing the elements of a unit yields a linear equation, and the set of all such equations forms a system of linear…

Let $L$ be an $n\times n$ array whose top left $r\times s$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We find necessary and sufficient conditions that ensure the…

This article could be called "theme and variations" on Cantor's celebrated diagonal argument. Given a square nxn tableau T=(a_i^j) on a finite alphabet A, let L be the set of its row-words. The permanent Perm(T) is the set of words…

A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats…

Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…

Two $n \times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that $L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$…

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a row and a column are (possibly…

The Super-Catalan numbers are a generalization of the Catalan numbers defined as $T(m,n) = \frac{(2m)!(2n)!}{2m!n!(m+n)!}$. It is an open problem to find a combinatorial interpretation for $T(m,n)$. We resolve this for $m=3,4$ using a…

By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study…

The usual, or type A_n, Tamari lattice is a partial order on T_n^A, the triangulations of an (n+3)-gon. We define a partial order on T_n^B, the set of centrally symmetric triangulations of a (2n+2)-gon. We show that it is a lattice, and…

A magic rectangle of order $m\times n$ with precisely $r$ filled cells in each row and precisely $s$ filled cells in each column, denoted $MR(m,n;r,s)$, is an arrangement of the numbers from 0 to $mr-1$ in an $m\times n$ array such that…

Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondr\'{a}k. Here, we extend the notion of isotopy, a permutation group action, from Latin squares to Latin…

A superpermutation is a sequence that contains every permutation of $n$ distinct symbols as a contiguous substring. For instance, a valid example for three symbols is a sequence that contains all six permutations. This paper introduces a…

Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define…