A Note on Generalized Narayana Numbers

TL;DR

This paper provides a straightforward proof for the enumeration of Dyck paths characterized by specific numbers of returns and peaks, expressed through generalized Narayana numbers, enhancing understanding of combinatorial path counting.

Contribution

It introduces a simple proof for the formula counting Dyck paths with given returns and peaks using generalized Narayana numbers.

Findings

Confirmed the enumeration formula for Dyck paths with specified returns and peaks

Connected Dyck path statistics to generalized Narayana numbers

Simplified the proof of a known combinatorial enumeration

Abstract

We give a simple proof that the the number of Dyck paths of semilength $n$ with $i$ returns to ground level and $j$ peaks is the generalized Narayana number $\frac{i}{n} \binom{n}{j} \binom{n - i - 1}{j - i}$.