منابع مشابه
Most Latin Squares Have Many Subsquares
A k_n Latin rectangle is a k_n matrix of entries from [1, 2, ..., n] such that no symbol occurs twice in any row or column. An intercalate is a 2_2 Latin subrectangle. Let N(R) be the number of intercalates in R, a randomly chosen k_n Latin rectangle. We obtain a number of results about the distribution of N(R) including its asymptotic expectation and a bound on the probability that N(R)=0. For...
متن کاملOverlapping latin subsquares and full products
We derive necessary and sufficient conditions for there to exist a latin square of order n containing two subsquares of order a and b that intersect in a subsquare of order c. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order n cannot have more than n m n h / m h subsquares of order m, where h = ⌈(m + 1)/2⌉. Indeed, the number of s...
متن کاملLifting Redundancy from Latin Squares to Pandiagonal Latin Squares
In the pandiagonal Latin Square problem, a square grid of size N needs to be filled with N types of objects, so that each column, row, and wrapped around diagonal (both up and down) contains an object of each type. This problem dates back to at least Euler. In its specification as a constraint satisfaction problem, one uses the all different constraint. The known redundancy result about all dif...
متن کاملLatin Squares: Transversals and counting of Latin squares
Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...
متن کاملAmalgamating infinite latin squares
A finite latin square is an n × n matrix whose entries are elements of the set {1, . . . , n} and no element is repeated in any row or column. Given equivalence relations on the set of rows, the set of columns, and the set of symbols, respectively, we can use these relations to identify equivalent rows, columns and symbols, and obtain an amalgamated latin square. There is a set of natural equat...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 1999
ISSN: 0097-3165
DOI: 10.1006/jcta.1998.2947