{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:31Z","timestamp":1753893811415,"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>The widely studied $q$-polynomial $f^{\\lambda}(q)$, which specializes when $q=1$ to $f^{\\lambda}$, the number of standard Young tableaux of shape $\\lambda$, has multiple combinatorial interpretations.  It represents the dimension of the unipotent representation $S_q^{\\lambda}$ of the finite general linear group $GL_n(q)$, it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index statistic on standard Young tableaux.  Similarly, the $q$-polynomial $g^{\\lambda}(q)$ has combinatorial interpretations as the $q$-multinomial coefficient, as the dimension of the permutation representation $M_q^{\\lambda}$ of the general linear group $GL_n(q)$, and as the generating function for both the inversion statistic and the charge statistic on permutations in $W_{\\lambda}$.  It is a well known result that for $\\lambda$ a partition of $n$, $dim(M_q^{\\lambda}) = \\Sigma_{\\mu} K_{\\mu \\lambda} dim(S_q^{\\mu})$, where the sum is over all partitions $\\mu$ of $n$ and where the Kostka number $K_{\\mu \\lambda}$ gives the number of semistandard Young tableaux of shape $\\mu$ and content $\\lambda$.  Thus $g^{\\lambda}(q) - f^{\\lambda}(q)$ is a $q$-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection $f$ from the set of standard Young tableaux of shape $\\lambda$, $SYT(\\lambda)$, to $W_{\\lambda}$ such that $maj(T) = ch(f(T))$ for $T \\in SYT(\\lambda)$.<\/jats:p>","DOI":"10.37236\/1942","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:09:47Z","timestamp":1578719387000},"source":"Crossref","is-referenced-by-count":1,"title":["A Relationship between the Major Index for Tableaux and the Charge Statistic for Permutations"],"prefix":"10.37236","volume":"12","author":[{"given":"Kendra","family":"Killpatrick","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2005,9,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r45\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:47:30Z","timestamp":1579322850000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1942","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,9,5]]},"article-number":"R45"}}