Proof of a Lattice Paths Conjecture Connected to the Tennis Ball Problem

The authors give a history of the so-called tennis ball problem, and discuss its relation to lattice path enumeration. We also prove a conjecture related to a solution of the symmetric case, namely when the number of balls removed each turn is exactly half the number inserted. Key words: Tennis Ball Problem, lattice path, Finite Operator Calculus 1 The Tennis Ball Problem-History The tennis ball problem was presented on pages 304 305 of the book by T.Tymoczko and J.Henle [12] in 1995. Their presentation deals with adding numbered books to a stack on a table, then removing some, in…nitely many times. Motivated by that presentation, Ralph P. Grimaldi and Joseph G. Moser deal with performing the process a …nite number of times [6]. Shanzhen Gao Email address: sgao2fau.edu (Shanzhen Gao). Preprint submitted to Journal of Statistical Planning and Inference20 October 2008 In their tennis ball problem, Colin L. Mallows and Lou Shapiro are given successive pairs of balls numbered (1; 2), (3; 4),... . At each stage they throw one ball out of the window. After n stages some set of n balls is on the lawn. They …nd a generating function and a explicit formula for the sequence 3; 23; 131; 664; 3166; 14545; 65187; 287060; 1247690,..., the n th term of which gives the sum over all possible arrangements of the total of the numbers on the balls on the lawn. They gave connections of the tennis ball problem with bicolored Motzkin paths and the ballot problem [9] . The (s; t) tennis ball problem goes as follows. At turn one, balls numbered 1 through s are thrown into a basket and a gnome in the basket picks t of them and throws them onto the lawn. At turn two, balls numbered s+1 through 2s are thrown into the basket and the gnome, now having 2s t balls to choose from, throws t of them onto the lawn. At the n-th turn balls (n 1)s+1 to sn go into the basket and the gnome throws out one the lawn t of the ns (n 1)t balls available to him. Question 1. Looking at the balls out on the lawn, how many di¤erent ball sequences, b1 < b2 < < btn are there? Question 2. What is the sum of the numbers of these ball sequences over all ball sequences of length tn? The most studied case is the (2,1) tennis ball problem. In this case Grimaldi andMoser showed that the answer to Question 1 isCn+1, whereCn = 2n n =(n+ 1) is the n-th Catalan number. Colin L. Mallows and Lou Shapiro answered the second question two years later, showing the result to be 2n + 5n+ 4 n+ 2 2n+ 1 n ! 2: In [10], both questions are answered for the cases (s; 1) and (4; 2). The methods involved are generating trees, recursion by level in these trees, Lagrange inversion, and the Riordan group.The (4; 2) case, treated in an appendix, is interesting because of connections to "Catalan trigonometry" [11] and to some problems involving minimally generated matroids and Tutte polynomials [1]. Mahendra Jani and Melkamu Zeleke give an alternative bijective proof of the generalized Tennis Ball Problem using k-trees. They also obtain a summation formula for all possible arrangements of balls out on the lawn using k-trees and lattice paths [7]. In the generalized tennis ball problem, sn balls are placed into a basket in n batches of s, and t balls are removed at every turn (0 < t < s). The sn balls are numbered, and we ask for the number of di¤erent sets of tn balls that are removed. With k = t and l = s t, this number is equal to