The cycle structure of two rows in a random Latin square

Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r, s be arbitrary distinct rows of L. We study the distribution of σr,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1− o(1) as n → ∞: (i) σr,s has more than (log n)1−c cycles, (ii) σr,s has fewer than 9√n cycles, (iii) L has fewer than 9 2n 5/2 intercalates (latin subsquares of order 2). We also show that the probability that σr,s is an even permutation lies in an interval [ 1 4 − o(1), 3 4 + o(1)] and the probability that it has a single cycle lies in [2n−2, 2n−2/3]. Indeed, we show that almost all derangements have similar probability (within a factor of n3/2) of occurring as σr,s as they do if chosen uniformly at random from among all derangements of {1, 2, . . . , n}. We conjecture that σr,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure of latin squares of orders n ≤ 11. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 33, 286–309, 2008