Dinv and Area

区域 $\defc\le\min(\area,\dinv)$ 中交换 area 和 dinv 的 Dyck 路径上的对合

An involution on Dyck paths in the region $\defc\le\min(\area,\dinv)$ that interchanges area and dinv

本文构造了满足 $\defc\le\min(\area,\dinv)$ 的 Dyck 路径上的显式对合，交换 area 和 dinv，基于新组合对象及双 Dyck 插入和 Garsia--Milne 原理。

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

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 $\vec{k}$-Dyck paths by: i) giv…

The $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q,t$-symmetry

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q,t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area,skips,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q,t$-symmetry of $(m…

An area-depth symmetric $q,t$-Catalan polynomial

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.

A description of the Zeta map on Dyck paths area sequences

We give a description of the well known Zeta map on Dyck paths which sends the dinv,area statistics to the area,bounce statistics. Our description uses Dyck paths area sequences and can be implemented easily.

Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.