On the Inverse Problem of the k-th Davenport Constants for Groups of Rank 2

Abstract

For a finite abelian group G and a positive integer k, let \(\textsf{D}_k(G)\) denote the smallest integer \(\ell \) such that each sequence over G of length at least \(\ell \) has k disjoint nontrivial zero-sum subsequences. It is known that \(\mathsf D_k(G)=n_1+kn_2-1\) if \(G\cong C_{n_1}\oplus C_{n_2}\) is a rank 2 group, where \(1<n_1\, | \,n_2\). We investigate the associated inverse problem for rank 2 groups, that is, characterizing the structure of zero-sum sequences of length \(\mathsf D_k(G)\) that can not be partitioned into \(k+1\) nontrivial zero-sum subsequences.