Some Variations on the Tennis Ball Problem

Abstract

Our study looks at Catalan, Motzkin, and Schr?der numbers. Catalan numbers are very well known mathematically and extensive research has been done highlighting their interesting properties. In contrast, Motzkin and Schroder numbers are not so well developed. We have been able to adapt certain techniques to different sequences. We have looked at variations of the Tennis Ball problem that generate Catalan and Motzkin numbers. We also investigated a recursive formula for counting Motzkin paths with flaws in terms of Motzkin numbers and came up with a combinatorial proof of Motzkin paths with 2 and 3 flaws in terms of normal Motzkin paths. Path and binary tree interpretations have also been looked at, leading to interesting results.