Bijections for Dyck paths with all peak heights of the same parity

TL;DR

This paper establishes bijective correspondences between specific classes of Dyck paths and well-known combinatorial sequences, revealing new connections between peak height parity and classical number sequences.

Contribution

It introduces bijections linking Dyck paths with all peaks at odd or even heights to Motzkin and Riordan numbers, respectively, highlighting novel combinatorial relationships.

Findings

Dyck paths with all peaks at odd height are counted by Motzkin numbers.

Dyck paths with all peaks at even height are counted by Riordan numbers.

Bijections provide combinatorial proofs of these counting results.

Abstract

We show bijectively that Dyck paths with all peaks at odd height are counted by the Motzkin numbers and Dyck paths with all peaks at even height are counted by the Riordan numbers.