On the number of fully weighted zero-sum subsequences

Abstract

Let [Formula: see text] be a finite additive abelian group with exponent [Formula: see text] and [Formula: see text] be a sequence of elements in [Formula: see text]. For any element [Formula: see text] of [Formula: see text] and [Formula: see text], let [Formula: see text] denote the number of subsequences [Formula: see text] of [Formula: see text] such that [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, we prove that [Formula: see text], when [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text], such that every sequence [Formula: see text] over [Formula: see text] of length at least [Formula: see text] has nonempty subsequence [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text]. Moreover, we classify the sequences such that [Formula: see text], where the exponent of [Formula: see text] is an odd number.