{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T09:56:27Z","timestamp":1772358987013,"version":"3.50.1"},"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>Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\\widetilde{H}_\\mu(\\mathbf{x};q,t)=\\widetilde{H}_{\\mu^\\ast}(\\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\\mathbf{x}$ for all shapes $\\mu$. We also provide a proof for the full relation in the case when $\\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.<\/jats:p>","DOI":"10.37236\/5350","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:05:25Z","timestamp":1578690325000},"source":"Crossref","is-referenced-by-count":3,"title":["A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials"],"prefix":"10.37236","volume":"23","author":[{"given":"Maria Monks","family":"Gillespie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2016,5,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p38\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p38\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:28:12Z","timestamp":1579238892000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i2p38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,27]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,3,31]]}},"URL":"https:\/\/doi.org\/10.37236\/5350","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,5,27]]},"article-number":"P2.38"}}