# Generalized Catalan Numbers

## Shelly Smith, GVSU Department of Mathematics

# Summer 2017 REU

There are numerous sets of combinatorial objects that are counted by the Catalan numbers $C(n)=\frac{1}{n+1}\binom{2n}{n}$, and many mathematicians have demonstrated bijections between these sets to prove that they have the same cardinality. The Catalan numbers can be generalized using parameters *p* and *r*, which correspond to the Raney numbers $R_{p,r}(n)=\frac{r}{np+r}\binom{np+r}{n}$, where $R_{2,1}(n)=C(n),$ but the research on these numbers and their applications is not as extensive.

## What will the project involve?

We will review existing research on sets of combinatorial objects that are counted by the Catalan numbers and explore combinatorial proofs that demonstrate a bijection between two such sets, proving that the sets have the same cardinality. We will then create sequences of generalized Catalan numbers and attempt to create new combinatorial sets counted by these sequences, then develop and prove bijections between these sets. Students will conduct research independently as well as in a team, and will work to improve their mathematical writing and presentation skills.

## Desirable experiences for applicants:

Applicants should have some experience in topics in discrete mathematics and counting techniques, including combinations and permutations. Applicants should also have experience writing mathematical proofs, and prior experience with proofs involving injective or surjective functions is preferred but not required.

## How to Apply:

For application information and instructions, please visit the GVSU Summer Mathematics REU home page.