On partitions of {1,…,2m + 1}{k} into differences d,…,d + m − 1: Extended Langford sequences of large defect

✍ Scribed by Vaclav Linek; Shai Mor

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 170 KB
• Volume: 12
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20001

✦ Synopsis

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.