{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T15:54:45Z","timestamp":1773158085294,"version":"3.50.1"},"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>We present two symmetric function operators $H_3^{qt}$ and $H_4^{qt}$ that have the property $H_{m}^{qt} H_{(2^a1^b)}(X;q,t) = H_{(m2^a1^b)}(X;q,t)$. These operators are generalizations of the analogous operator $H_2^{qt}$ and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, $a_{\\mu}(T)$ and $b_{\\mu}(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\\lambda$, $K_{\\lambda\\mu}(q,t) = \\sum_T  t^{a_{\\mu}(T)} q^{b_{\\mu}(T)}$ for the case when when $\\mu$ is  two columns or of the form $(32^a1^b)$ or $(42^a1^b)$. This provides proof of the positivity of the $(q,t)$-Kostka coefficients in the previously unknown cases of $K_{\\lambda (32^a1^b)}(q,t)$ and $K_{\\lambda (42^a1^b)}(q,t)$.  The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when $\\mu$ is two columns.<\/jats:p>","DOI":"10.37236\/1473","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T06:31:35Z","timestamp":1578637895000},"source":"Crossref","is-referenced-by-count":3,"title":["Positivity for Special Cases of $(q,t)$-Kostka Coefficients and Standard Tableaux Statistics"],"prefix":"10.37236","volume":"6","author":[{"given":"Mike","family":"Zabrocki","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1999,10,4]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v6i1r41\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v6i1r41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:32:07Z","timestamp":1579325527000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v6i1r41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,10,4]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1999,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1473","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,10,4]]},"article-number":"R41"}}