Dyck Numbers, I. Successor Function

TL;DR

This paper studies the minimal numbering of Dyck paths, introduces a successor function with closed formulas, and provides algorithms and hypotheses relating to Dyck numbers and related OEIS sequences.

Contribution

It presents a new successor function for Dyck numbers, derives closed formulas, and develops an algorithm and Python implementation for computing successors.

Findings

Closed formulas for the Dyck successor function are derived.

An algorithm and Python program for the Dyck successor function are provided.

A hypothesis relating Dyck numbers to OEIS sequence A001405 is formulated.

Abstract

Dyck paths are among the most heavily studied Catalan families. In the paper we are dealing with the minimal numbering of Dyck paths, with the resulting numbers, the terms of the OEIS sequence A036991, which we have called Dyck numbers. We consider the successor function on the Dyck numbers, closed formulas are obtained. In particular, the formula for the successor of the Mersenne number is calculated (since OEIS A000225 is a subset of A036991). We have obtained the corresponding algorithm of the Dyck successor function, DS-function, and a Python program has been compiled. At the end of the paper, the size of the A036991 ranges is associated with the terms of OEIS A001405. The corresponding hypothesis is formulated.