Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

Lattice path enumeration for semi-magic squares by Latin rectangles

Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this poset, and we completely describe the orbits, covering data, and maximal chains for squares of size 4, 5, and 6. The last item gives the number of Latin squares in these cases. To calculate efficiently for size 6, we in turn identify orbits with certain equivalence classes of hypergraphs.

Transversals in Latin Squares

A latin square of order $n$ is an $n\times n$ array of $n$ symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of $n$ entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogonal latin squares. Here we provide a brief survey of the literature on transversals. We cover (1) existence and enumeration results, (2) generalisations of transversals including partial tra…

拉丁方和交错符号超矩阵的Bruhat序

A Bruhat order for Latin squares and alternating sign hypermatrices

本文定义了拉丁方和交错符号超矩阵上的Bruhat序，通过角点和超矩阵的逐点支配关系刻画该序，并证明了角点和超矩阵构成分配格，但不同于二维情形，它不是拉丁方偏序集的Dedekind-MacNeille完备化。

New bounds for Ryser's conjecture and related problems

A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order $n\times n$ contains a transversal of order $n-1$. In this paper we prove the existence of a transversal of order $n-O(\log{n}/\log{\log{n}})$, i…

High Rate LDPC Codes from Difference Covering Arrays

This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from difference covering arrays. While the original construction by Gallagher was by randomly allocating bits in a sparse parity-check matrix, over the past 20 years researchers have used a variety of more structured approaches to construct these codes, with the more recent constructions of well-structured LDPC coming from balanced incomplete block designs (BIBDs) and from Latin squares over finite fields. However these constructions have suffered from the limited orders for which these designs exist. Her…

Edge covering pseudo-outerplanar graphs with forests

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerplanar graph, or two forests and a matching, or $\max\{Δ(G),4\}$ matchings, or $\max\{\lceilΔ(G)/2\rceil,3\}$ linear forests. These results generalize some ones on outerplanar graphs and $K_{2,3}$-minor-fre…