Enumeration of Fuss-Schr\"oder paths

TL;DR

This paper generalizes the enumeration of Fuss-Schr"oder paths of type , providing combinatorial interpretations and bijections with noncrossing partitions, extending previous work on small Schr"oder paths.

Contribution

It introduces a combinatorial interpretation for -Fuss-Schr"oder paths of type  and establishes bijections with specific noncrossing partitions, generalizing prior results.

Findings

Provides enumeration formulas for -Fuss-Schr"oder paths of type .

Establishes bijections with sparse noncrossing partitions.

Extends combinatorial interpretations from small Schr"oder paths to Fuss-Schr"oder paths.

Abstract

In this paper we enumerate the number of $(k, r)$-Fuss-Schr\"{o}der paths of type $\lambda$. Y. Park and S. Kim studied small Schr\"{o}der paths with type $\lambda$. Generalizing the results to small $(k, r)$-Fuss-Schr\"{o}der paths with type $\lambda$, we give a combinatorial interpretation for the number of small $(k, r)$-Fuss-Schr\"{o}der paths of type $\lambda$ by using Chung-Feller style. We also give two sets of sparse noncrossing partitions of $[2(k + 1)n + 1]$ and $[2(k + 1)n + 2]$ which are in bijection with the set of all small and large, respectively, $(k, r)$-Fuss-Schr\"{o}der paths of type $\lambda$.