Short note on the number of 1-ascents in dispersed dyck paths

TL;DR

This paper derives a closed-form formula for counting 1-ascents in all dispersed Dyck paths of a given length, advancing beyond previous implicit relations and asymptotic estimates.

Contribution

It provides the first explicit formula for the total number of 1-ascents in dispersed Dyck paths, improving understanding of their combinatorial structure.

Findings

Derived a closed-form formula for 1-ascents in dispersed Dyck paths

Extended previous work from implicit relations to explicit enumeration

Enhanced combinatorial understanding of dispersed Dyck paths

Abstract

A dispersed Dyck path (DDP) of length n is a lattice path on $N\times N$ from (0,0) to (n,0) in which the following steps are allowed: "up" (x, y) $\to$ (x+1, y+1); "down" (x, y) $\to$ (x+1, y-1); and "right" (x,0) $\to$ (x+1,0). An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length n, A191386 in Sloane's OEIS. Previously, only implicit generating function relations and asymptotics were known.