{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T19:49:22Z","timestamp":1789674562816,"version":"4.0.1"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11601448"],"award-info":[{"award-number":["11601448"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971395"],"award-info":[{"award-number":["11971395"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11901563"],"award-info":[{"award-number":["11901563"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the reciprocity $|M(G,|H|)|=|M(H,|G|)|$. Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version.<\/jats:p>","DOI":"10.1137\/20m1317529","type":"journal-article","created":{"date-parts":[[2021,5,20]],"date-time":"2021-05-20T11:53:17Z","timestamp":1621511597000},"page":"1077-1095","source":"Crossref","is-referenced-by-count":4,"title":["A Reciprocity on Finite Abelian Groups Involving Zero-Sum Sequences"],"prefix":"10.1137","volume":"35","author":[{"given":"Dongchun","family":"Han","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hanbin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,5,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02388522"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(82)90258-7"},{"key":"atypb3","first-page":"1","author":"Almkvist G.","year":"1978","journal-title":"Berlin"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(01)00343-0"},{"key":"atypb5","first-page":"202","author":"Armstrong D.","year":"2009","journal-title":"Mem. 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Geroldinger and F. Halter-Koch,\n                      Non-unique Factorizations. Algebraic, Combinatorial and Analytic Theory\n                      , Pure Appl. Math. (Boca Raton) 278, Chapman & Hall\/CRC, Boca Raton, FL, 2006.","DOI":"10.1201\/9781420003208"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-019-10079-0"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2012.07.002"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s10801-013-0443-z"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/6584"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022450120589"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2018.10.007"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.37236\/5734"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2012.10.006"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2008.05.003"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2011.07.003"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s11139-018-0020-0"},{"key":"atypb35","unstructured":"N. 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