Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

Abstract

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and each 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. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to 1/n. Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares T(n)≤nne−2n+o(n).

On the maximum number of Latin transversals

“…Improving on this, Taranenko [17] recently proved a better upper bound. Before stating her result, we shall need some notation.…”

Transversals in completely reducible multiary quasigroups and in multiary quasigroups of order 4

“…The problem of an upper bound on the number of transversals in latin squares of order q was posed by Wanless in Loops'03 and soon afterwards the first non-trivial asymptotic bound was proved in [5]. The bound was improved up to (1 + o(1)) q e 2 q in [10], and in [3] it was reproved by another technique. Moreover, in the latter paper it was proposed a probabilistic construction of latin squares that confirmed the exactness of this bound.…”

“…, a i n )} q i=1 which do not share a coordinate. As for latin hypercubes, papers [3] and [10] bound the number their transversals by (1 + o(1)) q n−1 e n q for large q.…”

Additive triples of bijections, or the toroidal semiqueens problem

“…Cooper and Kovalenko [CK96] showed that c 2 = e −0.08854 is acceptable, and this was later improved by Kovalenko [Kov96] to c 2 = 1/ √ 2 and by McKay, McLeod and Wanless [MMW06] to c 2 = 0.614. More recently, Taranenko [Tar15] proved that one can take c 2 = 1/e + o(1). Glebov and Luria [GL15] proved the same bound using a somewhat simpler method based on entropy.…”