{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T14:43:17Z","timestamp":1769006597592,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a finite abelian group of order $n$ and $\\mathcal M_G$ the Cayley table of $G$. Let $\\mathcal P(G)$ be the number of formally different monomials occurring in $\\mathsf {per}(\\mathcal M_G)$, the permanent of $\\mathcal M_G$. In this paper, for any finite abelian groups $G$ and $H$, we prove the following characterization $$\\mathcal P(G)=\\mathcal P(H)\\ \\Leftrightarrow\\ G\\cong H.$$ It follows that the group permanent determines the finite abelian group, which partially answers an open question of Donovan, Johnson and Wanless. In fact, $\\mathcal P(G)$ is closely related to zero-sum sequences over finite abelian groups and we shall prove the above characterization by studying a reciprocity of zero-sum sequences over finite abelian groups. As an application of our method, we show that $\\mathcal P(G)&gt;\\mathcal P(C_n)$ for any non-cyclic abelian group $G$ of order $n$ and thereby answer an open problem of Panyushev.<\/jats:p>","DOI":"10.37236\/13332","type":"journal-article","created":{"date-parts":[[2024,11,15]],"date-time":"2024-11-15T16:10:52Z","timestamp":1731687052000},"source":"Crossref","is-referenced-by-count":1,"title":["The Group Permanent Determines the Finite Abelian Group"],"prefix":"10.37236","volume":"31","author":[{"given":"Mao-sheng","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hanbin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,11,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i4p44\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i4p44\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,15]],"date-time":"2024-11-15T16:11:06Z","timestamp":1731687066000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i4p44"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,15]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,10,3]]}},"URL":"https:\/\/doi.org\/10.37236\/13332","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,15]]},"article-number":"P4.44"}}