منابع مشابه
Orthogonal Latin Rectangles
We use a greedy probabilistic method to prove that for every > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1− )n. That is, we show the existence of a second latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.
متن کاملCounting Three-Line Latin Rectangles
A k × n Latin rectangle is a k × n array of numbers such that (i) each row is a permutation of [n] = {1, 2, . . . , n} and (ii) each column contains distinct entries. If the first row is 12 · · ·n, the Latin rectangle is said to be reduced. Since the number k × n Latin rectangles is clearly n! times the number of reduced k× n Latin rectangles, we shall henceforth consider only reduced Latin rec...
متن کاملAsymptotic enumeration of Latin rectangles
A k X n Latin rectangle is a k X n matrix with entries from {1,2,.. . , n} such that no entry occurs more than once in any row or column. (Thus each row is a permutation of the integers 1,2,..., n.) Let L(k, n) be the number of k x n Latin rectangles. An outstanding problem is to determine the asymptotic value of L(k, n) as n —• oo, with k bounded by a suitable function of n. The first attack o...
متن کاملDivisors of the number of Latin rectangles
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) . Let Rk,n be the number of reduced k × n Latin rectangles and m = bn/2c. We prove several results giving divisors of Rk,n. For example, (k − 1)! divides Rk,n when k ≤ m and m! divides Rk,n when m < k ≤ n. We establish a recurrence which determ...
متن کاملCritical sets in back circulant Latin rectangles
A latin rectangle is an m × n array, m ≤ n, from the numbers 1, 2, . . . , n such that each of these numbers occur in each row and in each column at most once. A critical set in an m×n array is a set S of given entries, such that there exists a unique extension of S to a latin rectangle of size m×n. If we index the rows and columns of an m×n array, m ≤ n, by the sets M = {1, 2, . . . ,m} and N ...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1980
ISSN: 0012-365X
DOI: 10.1016/0012-365x(80)90136-3