Our object is to explore the ``stennis ball problem'' (at each round s balls are available and we play with one ball at a time). This is a natural generalization of the case s=2 considered by Mallows and Shapiro. We show how this generalization is connected with sary trees and employ the notion of generating trees to obtain a solution expressed in terms of generating functions. Then, we present a variation in which at each round we have 4 balls and play with 2 balls at a time. To solve this problem we use the concepts of Riordan arrays and stretched Riordan arrays, and a generalization of generating trees. This is a joint work by D. Merlini with D. G. Rogers, R. Sprugnoli and M. C. Verri.
The configuration after 4 rounds is (1, 2, 3, 6) for the first example and (1, 2, 3, 4) for the second example. In fact, for the (2,1)case, one has f_{1}=2, f_{2}=5, f_{3}=14, f_{4}=... do you guess what? There is indeed 42 different configurations (after 4 rounds), and if one adds all the sums, one gets S_{1}=1+2=3, S_{2}=(1+2)+(1+3)+(1+4)+(2+3)+(2+4)=23, S_{3}=131, S_{4}=664, ...
Turns Balls received Balls in the pocket Balls thrown away n=1 1 and 2 1 and 2 1 n=2 3 and 4 2, 3, and 4 3 n=3 5 and 6 2, 4, 5, and 6 2 n=4 7 and 8 4, 5, 6, 7, and 8 6 sum = 1+3+2+6= 12 Turns Balls received Balls in the pocket Balls thrown away n=1 1 and 2 1 and 2 2 n=2 3 and 4 1, 3, and 4 3 n=3 5 and 6 1, 4, 5, and 6 4 n=4 7 and 8 1, 5, 6, 7, and 8 1 sum = 2+3+4+1= 10
Figure 1: Two scenarios for the (s=2,t=1)tennis ball player.
This is the only case solved with t¹ 1. The general (s,t)tennis ball problem remains open.
Turns Balls received Balls in the pocket Balls thrown away n=1 1, 2, 3, 4 1, 2, 3, 4 2, 3, n=2 5, 6, 7, 8 1, 4, 5, 6, 7, 8 1, 7 n=3 9, 10, 11, 12 4, 5, 6, 8, 9, 10, 11, 12 10, 12 n=4 13, 14, 15, 16 4, 5, 6, 8, 9, 11, 13, 14, 15, 16 5, 16 2+3+1+7+10+12+5+16= 56
Figure 2: A scenario for the (4,2)tennis ball problem.
More generally, the rewriting rule {
Figure 3: The generating tree T for the (2,1)case and an isomorphic tree T^{~}.
root: (1) 
rule: (k) ® (1)...(k+s2)...(k+s1) 
T_{n}= 

and T(z)=1+zT(z)^{s} . 
S_{n1} = 


 




. 
S_{n}=A_{n} 

T_{n+1}. 
A(z)= 

+T'(z). 
2n 
n 
2n+1 
n 
M_{m}^{[n+1]}= 

(mr1)M_{r}^{[n]}. 
n/k  1  2  3  4  5  6  7  8  9 
0  1  
1  3  2  1  
2  22  16  10  4  1  
3  211  158  105  52  21  6  1  
4  2306  1752  1198  644  301  116  36  8  1 
d_{n+1,k+1}= 

a_{j} d_{n,k+j} for all n and k in N. 
d_{n,k}=[z^{n}] g(z)  (  zh(z)  )  ^{k} where h(z)=A  (  zh(z)  )  . 
n/k  0  1  2  3  4  5  6  7  
0  1  
1  0  1  
2  1  1  1  
3  0  3  2  1  
4  6  6  6  3  1  
5  0  22  16  10  4  1  
6  53  53  53  31  15  5  1 
2n+4 
n+2 
n+3 
(n+3)/2 
S_{n}= 


µ_{r}^{[2n2h]} w_{r}^{[2h]}= 


µ_{r}^{[2n2h]}w_{r}^{[2h]}. 
S(z)= 


(  µ_{r}(z)+µ_{r}(z)  )  (  w_{r}(z)+w_{r}(z)  )  =12z^{2} +284z^{4}+5436z^{6}+96768z^{8}+O(z^{10}). 
http://www.dsi.unifi.it/~merlini/Publications.html
.This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.