Alternating parity weak sequencing

Abstract

AbstractA subset of a group is ‐weakly sequenceable if there is an ordering of its elements such that the partial sums , given by and for , satisfy whenever and . By Costa et al., it was proved that if the order of a group is then all sufficiently large subsets of the nonidentity elements are ‐weakly sequenceable when is prime, and . Inspired by this result, we show that, if is the semidirect product of and and the subset is balanced, then admits, regardless of its size, an alternating parity ‐weak sequencing whenever is prime and . A subset of 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 that are semidirect products of a generic (nonnecessarily abelian) group and , that all sufficiently large balanced subsets of the nonidentity elements admit an alternating parity ‐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 of a group is large enough and if does not contain 0, then is ‐weakly sequenceable.