Most Latin Squares Have Many Subsquares

Ak×nLatin rectangle is ak×nmatrix of entries from {1, 2, …, n} such that no symbol occurs twice in any row or column. An intercalate is a 2×2 Latin sub-rectangle. LetN(R) be the number of intercalates inR, a randomly chosenk×nLatin rectangle. We obtain a number of results about the distribution ofN(R) including its asymptotic expectation and a bound on the probability thatN(R)=0. Forε>0 we prove most Latin squares of ordernhaveN(R)⩾n3/2−ε. We also provide data from a computer enumeration of Latin rectangles for smallk, n.