Hooked extended Langford sequences of small and large defects

Abstract

It is shown that for m = 2d +5, 2d+6, 2d+7 and d ≥ 1, the set {1, …, 2m + 1} − {k} can be partitioned into differences d, d + 1, …, d + m − 1 whenever (m, k) ≡ (0, 1), (1, d), (2, 0), (3, d+1) (mod (4, 2)) and 1 ≤ k ≤ 2m+1.

It is also shown that for m = 2d + 5, 2d + 6, 2d + 7, and d ≥ 1, the set {1, …, 2m + 2} − {k, 2m + 1} can be partitioned into differences d, d + 1, … …, d + m − 1 whenever (m, k) ≡ (0, 0), (1, d + 1), (2, 1), (3, d) (mod (4, 2)) and k ≥ m + 2.

These partitions are used to show that if m ≥ 8d + 3, then the set {1, … …, 2m+2}−{k, 2m+1} can be partitioned into differences d, d+1, …, d+m−1 whenever (m, k) ≡ (0, 0), (1, d+1), (2, 1), (3, d) (mod (4, 2)).

A list of values m, d that are open for the existence of these partitions (which are equivalent to the existence of Langford and hooked Langford sequences) is given in the conclusion.