{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T05:19:46Z","timestamp":1775711986568,"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>Garsia and Haiman (J. Algebraic. Combin.  $\\bf5$ $(1996)$, $191-244$) conjectured that a certain sum $C_n(q,t)$ of rational functions in $q,t$ reduces to a polynomial in $q,t$ with nonnegative integral coefficients.  Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. ${\\bf98}$ $(2001)$, $4313-4316$) the refined conjecture $C_n(q,t) = \\sum q^{{\\rm area}}t^{{\\rm bounce}}$.  Here the sum is over all Catalan lattice paths and ${\\rm area}$ and ${\\rm bounce}$ have simple descriptions in terms of the path.  In this article we give an extension of $({\\rm area},{\\rm bounce})$ to Schr\u00f6der lattice paths, and introduce polynomials defined by summing $q^{{\\rm area}}t^{{\\rm bounce}}$ over certain sets of Schr\u00f6der paths.  We derive recurrences and special values for these polynomials, and conjecture they are symmetric in $q,t$.  We also describe a much stronger conjecture involving rational functions in $q,t$ and the $\\nabla$ operator from the theory of Macdonald symmetric functions.<\/jats:p>","DOI":"10.37236\/1709","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:29:23Z","timestamp":1578709763000},"source":"Crossref","is-referenced-by-count":14,"title":["A Schr\u00f6der Generalization of Haglund's Statistic on Catalan Paths"],"prefix":"10.37236","volume":"10","author":[{"given":"E. S.","family":"Egge","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Haglund","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K.","family":"Killpatrick","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D.","family":"Kremer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2003,4,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:08:19Z","timestamp":1579324099000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,4,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1709","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,4,23]]},"article-number":"R16"}}