An Application to Fuss-Catalan Numbers

Year 2025, Volume: 13 Issue: 1, 12 - 20

Cemal Çiçek

https://doi.org/10.36753/mathenot.1560582

Abstract

This paper investigated how many different ways an $m$-rung staircase can be climbed within certain rules. It was observed that the climbing numbers of the stairs are related to 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 regarding the climbing numbers were obtained. 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 some Fuss-Catalan numbers.

Keywords

Lattice Path, Stairs, Catalan Numbers, Fuss-Catalan Numbers, Climbing

Project Number

FBA-2023-38876

References

• Tucker, A.: Applied Combinatorics. JohnWiley and Sons. New York, 2002.
• Vilenkin, N. Ya.: Combinatorics. Akademik Press. New York and London, 1971.
• Selimi, A., Saracevic, M.: Catalan numbers and applications. Vision International Scientific Journal. 4(1), 99-114 (2019).
• Koshy, T.: Catalan Numbers with Applications. Oxford University Press. New York, 2009.
• Stanley, R. P.: Catalan Numbers, Cambridge University Press. 2015.
• Roman, S.: An Introduction to Catalan Numbers. Springer. Switzerland, 2015.
• Aval, J.-C.: Multivariate Fuss-Catalan numbers. Discrete Mathematics. 308, 4660-4669 (2008).
• Alexandersson, P., Fufa, S. A., Getachew, F., Qiu, D.: Pattern-Avoıdance and Fuss–Catalan numbers. Journal of Integer Sequences. 26, 1-27 (2023).
• Berman, G., Fryer, K. D.: Introduction to Combinatorics. Akademik Press. New York and London, 1970.
• Özkan, E., Dervisoglu, S., Catalan numbers and modular arithmetic. Igdir University Journal of the Institute of Science and Technology. 2(4), 63-66 (2012).
• Simion, R., Schmidt, F. W.: Restricted Permutations, European Journal of Combinatorics. 6, 383– 406 (1985).
• Chou, W.-S., He, T.-X., Shiue, P.J.-S.: On the primality of the generalized Fuss-Catalan numbers. Journal of Integer Sequences, 21 , 1-8 (2018).