(i) σ_{r,s} has more than (log n)^{1-c} cycles,

(ii) σ_{r,s} has fewer than 9√n cycles,

(iii) L has fewer than (9/2)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 [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
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.

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Last modified: Mon Sep 18 15:29:26 EST 2006