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The Number of Transversals in a Latin Square

A latin square of order n is an n × n array of n symbols in
which each symbol occurs exactly once in each row and column. A
transversal is a set of n entries, one selected from each row and
each column of a latin square of order n such that no two entries contain
the same symbol. Define T(n) to be the maximum number of
transversals over all latin squares of order n. We show that b^{n} <
T(n) < c^{n}n!√n for n≥5 where b≈1.719 and
c≈0.614. A corollary of this result is an upper bound on the
number of placements of n non-attacking queens on an n × n
toroidal chess board.
Some divisibility properties of the number of transversals in latin squares
based on finite groups are established. We also provide data from a
computer enumeration of transversals in all latin squares of order at most 9,
all groups of order at most 23 and all possible turn-squares of order
14.

Various data collected during the writing of this paper can be found at
http://users.m
onash.edu.au/~iwanless/data/transversals/.

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