A set of necessary conditions for the existence of a partition of {1,... ,2m-1, L} into differences d, d +1,... ,d + m- 1 is (m, L)≡ (0,0), (1,d + 1), (2, 1), (3, d) (mod (4, 2)) and m ≥ 2d-2 or m = 1. If m = 2d - 2 then L = 5d - 5, if m = 2d - 1 then 4d - 2 ≤ L ≤ 6d -4 and if m ≥ 2d then 2m ≤ L ≤ 3m + d -2. Similar conditions for the partition of {1,... , 2m, L} \{2} into differences d, d + 1.....d+m-1are (m, L) ≡ (0,0), (1, d+1), (2,1), (3,d) (mod (4, 2)), (d, m, L) ≠ (1,1,4), (2,3,8) and m ≥ 2d - 2, m = 1 or (d, m, L) = (3, 2, 7), (3, 3, 9). If m = 2d-2 then L = 5d-5, 5d-3, if m = 2d-1 then 4d-1 ≤ L ≤ 6d-2 and if m ≥ 2d then 2m+1 ≤ L ≤ 3m+d-1. It is shown that for many cases when the necessary conditions hold, the set {1,....2m- 1, L} and {1.....2m- 1, L}\{2} can be so partitioned. These partitions exist for all the minimum and maximum L, when d ≤ 3, when m = 1 and when m ≥ 8d -3 (m ≥ 8d + 4 in the hooked case). The constructions given fully solve the existence of these partitions if the necessary conditions for the existence of extended and hooked extended Langford sequences are sufficient.

Journal of Combinatorial Mathematics and Combinatorial Computing
School of Mathematics and Statistics

Mor, S. (Shai), & Stevens, B. (2010). On partitions of {1,.., 2m + 1, l} into differences d,.., d + m: Stretched langford sequences. Journal of Combinatorial Mathematics and Combinatorial Computing, 73, 207–221.