On partitions of 1, …, 2m+ 1\k into differencesd, …, d+m− 1: Extended Langford sequences of large defect

Abstract It is shown that for m = 2 d − 1, 2 d, 2 d + 1, and d ≥ 1, the set 1, 2, …, 2 m + 2, − 2, k 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 (d, m, k) ≠ (1, 1, 3), (2, 3, 7) (where (x, y) ≡ (u, ν) mod (m, n) iff x ≡ u (mod m) and y ≡ ν (mod n) ). It is also shown that if m ≥ 2 d − 1 and m ∉ 2 d + 2, 8 d − 5, then the set 1, 2, …, 2 m + 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). Finally, for d = 4 we obtain a complete result for when 1, …, 2 m + 1 − k can be partitioned into differences 4, 5, …, m + 3. © 2004 Wiley Periodicals, Inc.