Divisors of the number of Latin rectangles

Abstract

A k × n Latin rectangle on the symbols {1, 2, . . . ,n} is called reduced if the first row is (1, 2, . . . ,n) and the first column is (1, 2, . . . ,k) T . Let R k,n be the number of reduced k × n Latin rectangles and m = n/2 . We prove several results giving divisors of R k,n . For example, (k − 1)! divides R k,n when k m and m! divides R k,n when m < k n. We establish a recurrence which determines the congruence class of R k,n (mod t) for a range of different t. We use this to show that R k,n ≡ ((−1) k−1 (k − 1)!) n−1 (mod n). In particular, this means that if n is prime, then R k,n ≡ 1 (mod n) for 1 k n and if n is composite then R k,n ≡ 0 (mod n) if and only if k is larger than the greatest prime divisor of n.

Hownotto prove the Alon-Tarsi conjecture

“…We begin by specializing the proof template in [23] to be applicable to Latin squares of a given sign. Let C be the set of all reduced Latin squares of order n. Consider a group G of isotopisms that acts on C. Suppose that G acts on a set A where {L ∈ C : | Atp(L) ∩ G| > 1} ⊆ A ⊆ C. Unless otherwise specified, we will assume that…”

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

“…divides R n . Theorem 2.2 implies that R n ≡ 0 (mod p) for all primes p m. In [2,24] we show that R p ≡ 1 (mod p) for prime p. The following theorem gives the value of R n (mod p) for primes p in the range m < p < n. Theorem 2.3. Let p be a prime such that n k p + 1.…”

“…We know that R n ≡ 0 (mod p) when n 2p by Theorem 2.2 and that R p ≡ 1 (mod p) by [2,24]. We now also know the value of R n (mod p) when p < n < 2p by Theorem 2.3.…”

“…In [23], we showed that R n+1 ≡ z n ≡ 0 (mod n) for composite n. Further congruences for the numbers of Latin squares and Latin rectangles were given in [2,22,24,25] (see [20] for a survey). Table 2 lists some values of χ(p, d) when p is a small odd prime number.…”

“…In [24], the authors proved various divisibility properties for the number of Latin rectangles. This article aims to extend those results through the study of partial orthomorphisms.…”