A Variation on the Tennis Ball Problem

03/25/2008 - 12:15pm
03/25/2008 - 1:10pm
Naiomi Cameron (Lewis and Clark College)

We consider a variation on the Tennis Ball Problem studied by Mallows-Shapiro and Merlini, et al. The $ s $-Tennis Ball problem is the following: At first turn, you are given $ s $ balls labeled $ 1,2,\dots,s $, where $ s $ is a fixed positive integer. You toss one of them out of the window onto the lawn. At the second turn, balls numbered $ s+1,s+2,\ldots,2s $ are given to you and now you toss any of the $ 2s-1 $ remaining balls onto the lawn. This continues for $ n $ turns. We consider two questions. First, how many different combinations of balls on the lawn are possible after $ n $ turns? Second, what is the sum of the labels of the balls on the lawn, over all distinct possibilities, after $ n $ turns? In the case where $ s=2, $ the solution to the original problem is the well known Catalan numbers. The variations discussed in this talk yield the Motzkin numbers and other related sequences.

Millikan 208, Pomona College
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