2010-05-01

# On partitions of {1,..., 2m + 1, l} into differences d,..., d + m: Stretched langford sequences

## Publication

### Publication

*Journal of Combinatorial Mathematics and Combinatorial Computing , Volume 73 p. 207- 221*

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.

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Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |

Citation |
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. |