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