Symmetry of the refined q,t-Catalan polynomials for k-Dyck paths

2025/10/09 by Qu, Menghao, Zhang, Yingrui
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.08196

Abstract

Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are q,t-symmetric by constructing an involution on plane trees. They also provided a new formula for the original q,t-Catalan polynomials C_n(q,t). We observe that depth is a slight modification of bounce, which was defined by the filling algorithm and ranking algorithm of Xin and the second author in their study of k-Dyck paths. In this article, we generalize depth of classical Dyck paths to the case of k-Dyck paths and prove q,t-symmetry of the pair of statistics (area, depth) for K-Dyck paths. We provide an alternative description of the higher q,t-Catalan polynomials C_n^(k)(q,t).

Citations

• Polyhedral geometry of refined q,t-Catalan numbers
• Shuffle theorems and sandpiles
• The q,t-symmetry of the generalized q,t-Catalan number C_(k1,k2,k3)(q,t) and C_(k,k,k,k)(q,t)
• On the Sweep Map for k-Dyck Paths
• On the Sweep Map for Fuss Rational Dyck Paths
• A proof of the shuffle conjecture
• Sweep maps: A continuous family of sorting algorithms
• A Fast Algorithm for MacMahon's Partition Analysis
• A Combinatorial Formula for the Character of the Diagonal Coinvariants
• MacMahon’s Partition Analysis: The Omega Package

Related

• Rational Catalan polynomials and rank words
• The (m,n)-rational q, t-Catalan polynomials for m=3 and their q,t-symmetry
• Dinv, Area, and Bounce for k-Dyck paths
• Polyhedral geometry of refined q,t-Catalan numbers
• Counting Dyck paths by area and rank