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 σ_r,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) σ_r,s has more than (log n)^1?c cycles, (ii) σ_r,s has fewer than 9 cycles, (iii) L has fewer than n^5/2 intercalates (latin subsquares of order 2). We also show that the probability that σ_r,s is an even permutation lies in an interval 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 n^3/2) of occurring as σ_r,s as they do if chosen uniformly at random from among all derangements of {1,2,…,n}. We conjecture that σ_r,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., 2008