The Generalized Schroder Theory

While the standard Catalan and Schroder 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 Schroder theory. We define $m$-Schroder paths, find the number of such paths from $(0,0)$ to $(mn, n)$, and obtain some other results on the $m$-Schroder paths and $m$-Schroder words. Hoping to generalize classical $q$-analogue results of the ordinary Catalan and Schroder numbers, such as in the works of Furlinger and Hofbauer, Cigler, and Bonin, Shapiro and Simion, we derive a $q$-identity which would welcome a combinatorial interpretation. Finally, we introduce the ($q, t$)-$m$-Schroder polynomial through "$m$-parking functions" and relate it to the $m$-Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov.