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 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) r,s has more than (log n)1-c cycles, (ii) r,s has fewer than 9 n cycles, (iii) L has fewer than 9 2 n5/2 intercalates (latin subsquares of order 2). We also show that the probability that r,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 r,s as they do if chosen uniformly at random from among all derangements of {1, 2, . . . , n}. We conjecture that r,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure ...

• Type: Journal
• Year: 2008
• Where: RSA
• Authors: Nicholas J. Cavenagh, Catherine S. Greenhill, Ian M. Wanless
• Keywords: Arbitrary Distinct Rows, Latin Squares, Random, RSA 2008