On the number of additive permutations and Skolem-type sequences

Abstract

Cavenagh and Wanless recently proved that, for sufficiently large odd n , the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246) n . We adapt their proof to show that for sufficiently large t the number of additive permutations on [− t , t ] is greater than (3.246) 2 t + 1 and we go on to derive some much improved lower bounds on the numbers of Skolem-type sequences. For example, it is shown that for sufficiently large t ≡ 0 or 3 (mod 4) , the number of split Skolem sequences of order n = 7 t + 3 is greater than (3.246) 6 t + 3 . This compares with the previous best bound of 2 ⌊ n /3⌋ .