An Application to Fuss-Catalan Numbers

In this paper, it was investigated how many different ways an $m$-rung stairs can be climbed within certain rules. It was observed that the climbing numbers of the stairs have relations with the Catalan numbers. The combinatorics problem discussed in this article is different from the ones done so far and is related not only to Catalan numbers but also to some Fuss-Catalan numbers. Some results were obtained regarding the climbing numbers. It was observed that with the initial ascent being fixed, the climbing numbers of stairs with $m, m+1, m+2, m+3$, ... rungs, where $m>1$ is an integer, are related to respectively the some Fuss-Catalan numbers.