{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T02:49:24Z","timestamp":1774320564519,"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>While the standard Catalan and Schr\u00f6der  theories  both have been extensively studied,  people have only begun to investigate higher dimensional versions of the Catalan number (see, say, the 1991 paper of Hilton and Pedersen, and the 1996 paper of Garsia and Haiman). In this paper, we study a yet more general case, the higher dimensional Schr\u00f6der theory. We define $m$-Schr\u00f6der paths, find the number of such paths from $(0,0)$ to $(mn, n)$, and obtain some other results on the $m$-Schr\u00f6der paths and $m$-Schr\u00f6der words. Hoping to generalize classical $q$-analogue results of the ordinary Catalan and Schr\u00f6der numbers, such as in the works of F\u00fcrlinger and Hofbauer, Cigler, and Bonin, Shapiro and Simion, we derive a $q$-identity which would welcome a combinatorial interpretation. Finally, we introduce the ($q, t$)-$m$-Schr\u00f6der polynomial through \"$m$-parking functions\" and relate it to the $m$-Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov.<\/jats:p>","DOI":"10.37236\/1950","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:09:12Z","timestamp":1578719352000},"source":"Crossref","is-referenced-by-count":9,"title":["The Generalized  Schr\u00f6der Theory"],"prefix":"10.37236","volume":"12","author":[{"given":"Chunwei","family":"Song","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,10,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r53\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:40:30Z","timestamp":1579322430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,10,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1950","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,10,20]]},"article-number":"R53"}}