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 =>0 we prove most Latin squares of order n have N(R) n . We also provide data from a computer enumeration of Latin rectangles for small k, n.