Refinements of Lattice paths with flaws

TL;DR

This paper refines the enumeration of Dyck paths with flaws by analyzing parameters like peaks, valleys, and double ascents, establishing reciprocity theorems and Chung-Feller properties, and connecting to Narayana numbers.

Contribution

It introduces new refinements and Chung-Feller type theorems for Dyck paths with flaws based on multiple parameters, expanding classical combinatorial results.

Findings

Derived reciprocity theorem for polynomial P_{n,m}(x).

Established Chung-Feller properties for sums involving p_{n,m,k}.

Connected the enumeration of Dyck paths with double ascents to Narayana numbers.

Abstract

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ valleys (resp. double descents). Some similar results are derived.