{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:47Z","timestamp":1753893827103,"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>Foata and Zeilberger defined the graphical major index, $\\mathrm{maj}_U$, and the graphical inversion index, $\\mathrm{inv}_U$, for words over the alphabet $\\{1, 2, \\dots, n\\}$. These statistics are a generalization of the classical permutation statistics $\\mathrm{maj}$ and $\\mathrm{inv}$ indexed by directed graphs $U$. They showed that $\\mathrm{maj}_U$ and $\\mathrm{inv}_U$ are equidistributed over all rearrangement classes if and only if $U$ is bipartitional. In this paper we strengthen their result by showing that if $\\mathrm{maj}_U$ and $\\mathrm{inv}_U$ are equidistributed on a single rearrangement class then $U$ is essentially bipartitional. Moreover, we define a graphical sorting index, $\\mathrm{sor}_U$, which generalizes the sorting index of a permutation. We then characterize the graphs $U$ for which $\\mathrm{sor}_U$ is equidistributed with $\\mathrm{inv}_U$ and $\\mathrm{maj}_U$ on a single rearrangement class.\u00a0<\/jats:p>","DOI":"10.37236\/6263","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:42:39Z","timestamp":1578652959000},"source":"Crossref","is-referenced-by-count":0,"title":["Graphical Mahonian Statistics on Words"],"prefix":"10.37236","volume":"25","author":[{"given":"Amy","family":"Grady","sequence":"first","affiliation":[]},{"given":"Svetlana","family":"Poznanovi\u0107","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,1,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:41:46Z","timestamp":1579218106000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i1p1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,3]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/6263","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,1,3]]},"article-number":"P1.1"}}