Search

A subset S $S$ of a group (G,+) $(G,+)$ is t $t$‐weakly sequenceable if there is an ordering (y1,...,yk) $({y}_{1},{\rm{\ldots }},{y}_{k})$ of its elements such that the partial sums s0,s1,...,sk ${s}_{0},{s}_{1},{\rm{\ldots }},{s}_{k}$, given by s0=0 ${s}_{0}=0$ and si=∑j=1iyj ${s}_{i}={\sum }_{j=1}^{i}{y}_{j}$ for 1≤i≤k $1\le i\le k$, satisfy si≠sj ${s}_{i}\ne {s}_{j}$ whenever and 1≤∣i−j∣≤t $1\le | i-j| \le t$. By Costa et al., it was proved that if the order of a group is pe $pe$ then all sufficiently large subsets of the nonidentity elements are t $t$‐weakly sequenceable when p>3 $p\gt 3$ is prime, e≤3 $e\le 3$ and t≤6 $t\le 6$. Inspired by this result, we show that, if G $G$ is the semidirect product of Zp ${{\mathbb{Z}}}_{p}$ and Z2 ${{\mathbb{Z}}}_{2}$ and the subset S $S$ is balanced, then S $S$ admits, regardless of its size, an alternating parityt $t$‐weak sequencing whenever p>3 $p\gt 3$ is prime and t≤8 $t\le 8$. A subset of G $G$ is balanced if it contains the same number of even elements and odd elements and an alternating parity ordering alternates even and odd elements. Then using a hybrid approach that combines both Ramsey theory and the probabilistic method we also prove, for groups G $G$ that are semidirect products of a generic (nonnecessarily abelian) group N $N$ and Z2 ${{\mathbb{Z}}}_{2}$, that all sufficiently large balanced subsets of the nonidentity elements admit an alternating parity t $t$‐weak sequencing. The same procedure works also for studying the weak sequenceability for generic sufficiently large (not necessarily balanced) sets. Here we have been able to prove that, if the size of a subset S $S$ of a group G $G$ is large enough and if S $S$ does not contain 0, then S $S$ is t $t$‐weakly sequenceable. [ABSTRACT FROM AUTHOR]