{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:21:16Z","timestamp":1759335676307,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For $n\\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\\oplus C_{mn}$ with $n\\geq 2$ and $m\\geq 1$, and let $k\\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to characterize those sequences with maximal length $mn+n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $mn+n-1-k$. For $k\\leq 1$, this is the inverse question for the Davenport Constant. For $k=n-1$, this is the inverse question for the $\\eta(G)$ invariant concerning short zero-sum subsequences. For $C_n\\oplus C_n$ and $k\\in [2,n-2]$, with $n\\geq 5$ prime, it was conjectured in a paper of Grynkiewicz, Wang and Zhao that they must have the form $S=e_1^{[n-1]}\\boldsymbol{\\cdot} e_2^{[n-1]}\\boldsymbol{\\cdot} (e_1+e_2)^{[k]}$ for some basis $(e_1,e_2)$, with the conjecture established in many cases and later extended to composite moduli $n$. In this paper, we focus on the case $m\\geq 2$. Assuming the conjectured structure holds for $k\\in [2,n-2]$ in $C_n\\oplus C_n$, we characterize the structure of all sequences of maximal length $mn+n-2+k$ in $C_n\\oplus C_{mn}$ that fail to contain a nontrivial zero-sum of length at most $mn+n-1-k$, showing they must either have the form $S=e_1^{[n-1]}\\boldsymbol{\\cdot} e_2^{[sn-1]}\\boldsymbol{\\cdot} (e_1+e_2)^{[(m-s)n+k]}$ for some $s\\in [1,m]$ and basis $(e_1,e_2)$ with $\\mathsf{ord}(e_2)=mn$, or else have the form $S=g_1^{[n-1]}\\boldsymbol{\\cdot} g_2^{[n-1]}\\boldsymbol{\\cdot} (g_1+g_2)^{[(m-1)n+k]}$ for some generating set $\\{g_1,g_2\\}$ with $\\mathsf{ord}(g_1+g_2)=mn$. In view of prior work, this reduces the structural characterization for a general rank two abelian group to the case $C_p\\oplus C_p$ with $p$ prime. Additionally, we give a new proof of the precise structure in the case $k=n-1$ for $m=1$. Combined with known results, our results unconditionally establish the structure of extremal sequences in $G=C_n\\oplus C_{mn}$ in many cases, including when $n$ is only divisible by primes at most $7$, when $n\\geq 2$ is a prime power and $k\\leq \\frac{2n+1}{3}$, or when $n$ is composite and $k=n-d-1$ or $n-2d+1$ for a proper, nontrivial divisor $d\\mid n$.<\/jats:p>","DOI":"10.37236\/10921","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T17:25:13Z","timestamp":1657819513000},"source":"Crossref","is-referenced-by-count":2,"title":["A Multiplicative Property for Zero-Sums II"],"prefix":"10.37236","volume":"29","author":[{"given":"David J.","family":"Grynkiewicz","sequence":"first","affiliation":[]},{"given":"Chao","family":"Liu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2022,7,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i3p12\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i3p12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T17:25:14Z","timestamp":1657819514000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i3p12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,15]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/10921","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,7,15]]},"article-number":"P3.12"}}