A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

Abstract

A partial orthomorphism of Z n is an injective map σ : S → Z n such that S ⊆ Z n and σwhen p is a prime and n k p + 1. In particular, this enables us to calculate some previously unknown congruences for R n, n . We also develop techniques for computing ω(n, d) exactly. We show that for each a there exists μ a such that, on each congruence class modulo μ a , ω(n, n−a) is determined by a polynomial of degree 2a in n. We give these polynomials for 1 a 6, and find an asymptotic formula for ω(n, n − a) as n → ∞, for arbitrary fixed a.

Enumerating Partial Latin Rectangles

“…In this section, we describe an inclusion-exclusion method (based in turn on a graph theoretic approach) for finding formulas for the size of PLR(r, s, n; m), with m 1. To this end, we modify conveniently the method for enumerating partial orthomorphisms of finite cyclic groups given in [66]. At first glance, this may seem surprising as partial Latin rectangles and partial orthomorphisms are largely unrelated (unless we impose some symmetry, which we don't in the context of this section).…”

“…Symmetries of Latin squares and rectangles have been studied in a wide range of contexts, e.g., enumeration [57,65,66], subsquares [13,55], the Alon-Tarsi Conjecture [21,67], quasigroups and loops [8,44,45,54], special kinds of symmetries [15,33,42,72], and in their own right [9,14,24,25]. They are beginning to find applications in secret sharing schemes [23,71,75], erasure codes [76,68], and graph coloring games [5,4].…”

On Computing the Number of Latin Rectangles

“…This question, along with analogous questions for Latin rectangles, has been a hot research topic for Stones [35], and is linked to the autotopisms and automorphisms of Latin rectangles [4,39] (see also [12,38]), and orthomorphisms and partial orthomorphisms of finite cyclic groups [40,41]. Divisors for the number of even/odd Latin squares have been used in proving special cases of the Alon-Tarsi Conjecture [7] (see also [8,16,37,42]).…”

On the Number of Latin Rectangles

“…The enumeration of partial orthomorphisms is also linked to the value of R k,n [17]. We give new sufficient conditions for when a partial orthomorphism admits a completion to an orthomorphism and give a method for finding the number of partial orthomorphisms with |S| = a, for fixed a [17].…”