{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:49Z","timestamp":1753893829449,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableaux of various shapes $\\lambda$ and contents $\\mu$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableaux that \"standardize\" to a fixed semistandard Young tableau. For $m$-row shapes $\\lambda$ and standard content $\\mu$, the total number of $k$-inverted standard Young tableaux of shape $\\lambda$ is then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of \"returns to ground\" in those paths. In the rectangular specialization of $\\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\\lambda=2^n$.<\/jats:p>","DOI":"10.37236\/6376","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:55:11Z","timestamp":1578671711000},"source":"Crossref","is-referenced-by-count":0,"title":["Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers"],"prefix":"10.37236","volume":"24","author":[{"given":"Paul","family":"Drube","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,5,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p26\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:57:00Z","timestamp":1579237020000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i2p26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,19]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6376","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,5,19]]},"article-number":"P2.26"}}