Dinv, Area, and Bounce for $\vec{k}$-Dyck paths

Abstract

The well-known q, t-Catalan sequence has two combinatorial interpretations as weighted sums of ordinary Dyck paths: one is Haglund's area-bounce formula, and the other is Haiman's dinv-area formula. The zeta map was constructed to connect these two formulas: it is a bijection from ordinary Dyck paths to themselves, and it takes dinv to area, and area to bounce. Such a result was extended for k-Dyck paths by Loehr. The zeta map was extended by Armstrong-Loehr-Warrington for a very general class of paths.In this paper, We extend the dinv-area-bounce result for k-Dyck paths by: i) giving a geometric construction for the bounce statistic of a k-Dyck path, which includes the k-Dyck paths and ordinary Dyck paths as special cases; ii) giving a geometric interpretation of the dinv statistic of a k-Dyck path. Our bounce construction is inspired by Loehr's construction and Xin-Zhang's linear algorithm for inverting the sweep map on k-Dyck paths. Our dinv interpretation is inspired by Garsia-Xin's visual proof of dinv-to-area result on rational Dyck paths.

The $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{(k_1,k_2,k_3)}(q,t)$ and $C_{(k,k,k,k)}(q,t)$

“…Further information about q, t-Catalan sequence and related results can be found in [2,4,6]. In [11], we introduced q, t-Catalan numbers C λ (q, t) of type (a partition) λ as an extension of Haglund and Haiman's combinatorial formula for ordinary q, t-Catalan numbers. We also investigated the q, t-symmetry of C λ (q, t): The symmetry is easily proved when the length of λ is (λ) = 2; The symmetry may be proved by using MacMahon's partition analysis technique when (λ) = 3; No symmetry holds in general when (λ) ≥ 4.…”