Hooked extended Langford sequences of small and large defects

Rok, strany: 2014, 819 - 842

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 $(m,k)\equiv (0,1),(1,d),(2,0),(3,d+1)\pmod{(4,2)}$ 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 $(m,k)\equiv (0,0),(1,d+1),(2,1),(3,d)\pmod{(4,2)}$ 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 $(m,k)\equiv (0,0),(1,d+1),(2,1),(3,d)\pmod{(4,2)}$. 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.

ISO 690:

Mor, S., Linek, V. 2014. Hooked extended Langford sequences of small and large defects. In Mathematica Slovaca, vol. 64, no.4, pp. 819-842. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0242-6

APA:

Mor, S., Linek, V. (2014). Hooked extended Langford sequences of small and large defects. Mathematica Slovaca, 64(4), 819-842. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0242-6