Extensions of 2-rowed Latin rectangles
Here is some data about the number of ways to extend a 2-rowed Latin
rectangle to a Latin square. Any 2 by n Latin rectangle gives rise to
a derangement (a fixed-point-free permutation) of the set [1..n], and
the number of extensions depends only on the conjugacy class (that is,
the cycle type) of this derangement. The cycle type is specified by
the list of cycle lengths. The results for n=9 were found by Ian
Wanless.
n=4
[4] 2
[2,2] 4
n=5
[5] 36
[3,2] 24
n=6
[6] 4032
[4,2] 4224
[3,3] 4608
[2,2,2] 5376
n=7
[7] 6566400
[5,2] 6604800
[4,3] 6543360
[3,2,2] 6635520
n=8
[8] 181519810560
[6,2] 182125854720
[5,3] 181364244480
[4,4] 182052126720
[4,2,2] 183299604480
[3,3,2] 181813248000
[2,2,2,2] 186042286080
n=9
[9] 113959125225308160
[7,2] 114140503159603200
[6,3] 113970892709560320
[5,4] 113938545628938240
[5,2,2] 114303522444410880
[4,3,2] 114131854216396800
[3,3,3] 113995242201415680
[3,2,2,2] 114460947413729280
The obvious conjecture is that the ratio of the largest to the smallest
of these numbers tends to 1. This would have the consequence that the
group generated by the rows of a random normalised Latin square (first
row is the identity permutation) is the symmetric group with probability
tending to 1. One could also ask: for which cycle structures are the
largest and smallest values attained?
Peter Cameron, 23 May 2006