Most Latin squares have many subsquares

A k x n Latin rectangle is a k x n matrix of entries from {1, 2...., n} such that no symbol occurs twice in any row or column. An intercalate is a 2x2 Latin subrectangle. Let N(R) be the number of intercalates in R, a randomly chosen k x 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 epsilon>0 we prove most Latin squares of order n have N(R)greater than or equal to n(3/2-epsilon). We also provide data from a computer enumeration of Latin rectangles for small k,n. (C) 1999 Academic Press.