The Generalized Schröder Theory

Abstract

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

Rectangular Schroder Parking Functions Combinatorics

“…The aim of this work is to investigate properties of such paths embedded not into a square (n, n), but into a rectangle (m, n). Partial results are already known, in the case where m and n are coprime, in particular when m = rn + 1, which reduces to mn = n [23]. In the present article, (see Proposition 1) we obtain a generating series formula, in the general case (no coprimality required).…”

“…In particular, in view of (8), this covers the classical case (m = n) as well as the generalized version (m = rn) of [23]. One also deduces from Proposition 2 the following generalization of a result of Haglund [15].…”

“…Schröder polynomials. Although the general notion of (m, n)-Schröder paths considered here seems to be new, the special case of m = rn + 1 has been considered in [23]. The "classical case" corresponds to m = n. In this case, Schröder polynomials, given by the following formula…”

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity

“…Likewise, in presenting m-Dyck and m-Schröder paths, some authors [17,144] use an analogously turned picture (although for m = 1 the required linear transformation is no longer angle-preserving): the roles of up, down and long steps are again played by (0, 1), (1, 0) and (1,1), and now the path runs from (0, 0) to (n, mn) while staying in the region y ≥ mx. Alternatively, the roles of north and east can be reversed: then the path runs from (0, 0) to (mn, n) while staying in the region y ≤ x/m.…”

Enumeration on Row-Increasing Tableaux of Shape $2 \times n$

“…There is an obvious bijection between SSYTs in Inc k (2 × n) and small Schröder n-paths with k flat steps: read the numbers in a tableau from 1 to 2n − k in increasing order, if i appears only in row 1 (2), it corresponds to a U (D) step; if i appears in both rows, it corresponds to an F step. Increasing rectangular tableaux and m-Schröder paths are also studied in [5,8].…”