MIN-turns and MAX-turns in k-Dyck paths: a pure generating function approach

TL;DR

This paper introduces a generating function approach to analyze the levels of k-Dyck paths between specific up-steps, providing new insights into their structure without relying heavily on combinatorial arguments.

Contribution

It presents a novel generating function method to study max-terms and min-terms in k-Dyck paths, extending classical results with minimal combinatorial reasoning.

Findings

Derived explicit generating functions for max-terms and min-terms

Provided new enumerative formulas for levels in k-Dyck paths

Connected results to historical work by Rainer Kemp

Abstract

$k$-Dyck paths differ from ordinary Dyck paths by using an up-step of length $k$. We analyze at which level the path is after the $s$-th up-step and before the $(s+1)$st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago the terms \textsc{max}-terms and \textsc{min}-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.