{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T17:15:12Z","timestamp":1758474912548,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of the $2k$-th symmetrized moment $\\eta_{2k}(n)$ of ranks of partitions of $n$ in terms of $(k+1)$-marked Durfee symbols of $n$. In this paper, we consider the $k$-th symmetrized positive moment $\\bar{\\eta}_k(n)$ of ranks of partitions of $n$ which is defined as the truncated sum over positive ranks of partitions of $n$. As combinatorial interpretations of $\\bar{\\eta}_{2k}(n)$ and $\\bar{\\eta}_{2k-1}(n)$, we show that for given $k$ and $i$ with $1\\leq i\\leq k+1$, $\\bar{\\eta}_{2k-1}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being zero and $\\bar{\\eta}_{2k}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being positive. The interpretations of $\\bar{\\eta}_{2k-1}(n)$ and $\\bar{\\eta}_{2k}(n)$ are independent of $i$, and they imply the interpretation of $\\eta_{2k}(n)$ given by Andrews since $\\eta_{2k}(n)$ equals $\\bar{\\eta}_{2k-1}(n)$ plus twice of $\\bar{\\eta}_{2k}(n)$. Moreover, we obtain the generating functions for $\\bar{\\eta}_{2k}(n)$ and $\\bar{\\eta}_{2k-1}(n)$.<\/jats:p>","DOI":"10.37236\/3852","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:10:03Z","timestamp":1578705003000},"source":"Crossref","is-referenced-by-count":3,"title":["On the Positive Moments of Ranks of Partitions"],"prefix":"10.37236","volume":"21","author":[{"given":"William Y. C.","family":"Chen","sequence":"first","affiliation":[]},{"given":"Kathy Q.","family":"Ji","sequence":"additional","affiliation":[]},{"given":"Erin Y. Y.","family":"Shen","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2014,2,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:05:26Z","timestamp":1579259126000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i1p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/3852","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,2,7]]},"article-number":"P1.29"}}