Golliwog Numbers

Golliwog Numbers - lower left is G(1,1)
0	0	0	0	0	0	0	0	0	0	1	120543840
0	0	0	0	0	0	0	0	0	1	10628640	6668781166080
0	0	0	0	0	0	0	0	1	1026576	628308907776	303443622431870976
0	0	0	0	0	0	0	1	109584	7245893376	381495483224064	17810567950611972096
0	0	0	0	0	0	1	13068	104587344	673781602752	3878864920694016	21006340945438768128
0	0	0	0	0	1	1764	1942416	1744835904	1413470290176	1083688832185344	806595068762689536
0	0	0	0	1	274	48076	6998824	929081776	117550462624	14500866102976	1765130436471424
0	0	0	1	50	1660	46760	1217776	30480800	747497920	18139003520	437786795776
0	0	1	11	85	575	3661	22631	13745	833375	5019421	30174551
0	1	3	7	15	31	63	127	255	511	1023	2047
1	1	1	1	1	1	1	1	1	1	1	1

This differs from the table in [DKB] only by the correction of an obvious typo. We note that the non-zero entries in row k (from the bottom) have as rational generating function

                    k
                _________
                  |   |
                  |   |             1
                  |   |     _________________    
                  |   |              
                  |   |              k! 
                  |   |       1 -  ______ z  
                  |   |                        
                  |   |              p                    
                  
                   p=1

Pez probabilities

Table of golliwog Numbers - g(m,k) - G(m,k) divided by k!^m-k+1
0	0	0	0	0	0	0	0	0	0	0	0.000000002
0	0	0	0	0	0	0	0	0	0	0.000000025	0.000000076
0	0	0	0	0	0	0	0	0	0.00000028	0.0000008	0.0000014
0	0	0	0	0	0	0	0	0.000003	0.0000078	0.000013	0.0000175
0	0	0	0	0	0	0	0.000025	0.000067	0.00011	0.000144	0.000167
0	0	0	0	0	0	0.000198	0.000514	0.000827	0.00104	0.00119	0.00128
0	0	0	0	0	0.00139	0.00340	0.00520	0.00649	0.007305	0.00778	0.00804
0	0	0	0	0.00833	0.01903	0.027822	0.03375	0.03734	0.03937	0.04047	0.04105
0	0	0	0.04167	0.08681	0.12008	0.14094	0.15294	0.15950	0.16298	0.16479	0.16571
0	0	0.16667	0.30556	0.39352	0.44367	0.47081	0.48506	0.49242	0.49617	0.49807	0.49903
0	0.50000	0.75000	0.87500	0.93750	0.96875	0.98438	0.99219	0.99609	0.99805	0.99902	0.99951
1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0

This table is perhaps more fun to read than the more extensive Table 4.5.2 in [DKB]. In the context of the Pez problem we see, of course, that for N distinct flavors, the probability that there is a flavor missed by the sharer (or victim!) goes to zero as the number of dispensers increases (trace the probabilities up a diagonal), while if we consider the number of dispensers fixed at k, and the number of flavors (hence the height of the dispensers) to increase, the probability of missing a flavor [(k-1)!*g(m,k)] seems to approach 1 (This limit is easily verified using the iterated difference formula for g(m,k)).

Len Smiley

Last modified: Wed Nov 21 01:17:16 AKST 2001