Concrete mathematics : a foundation for computer science

8.5 HASHING 407
The probability that the list sizes will be nl , n2, . . . , n, is the multinomial
coefficient
(
n
nl,nz,...,n,
>
=
n!
nl!n2! . ..n.!
divided by mn; hence the pgf for B( hl , . . . , h,) is
B,(z) = = (
n1 ,n2 ,....n,>o
n, +n2 t...+n,=n
n
nl,nz,...,n,
>
J;‘)+(;‘)+-+(“jq m-n.
This sum looks a bit scary to inexperienced eyes, but our experiences in
Chapter 7 have taught us to recognize it as an m-fold convolution. Indeed, if
we consider the exponential super-generating function
G(w,z) = ~Bn,z,~,
n20
we can readily verify that G (w, z) is simply an mth power:
As a check, we can try setting z = 1; we get G(w, 1) = (ew)m, so the coefficient
of m”w”/n! is B, (1) = 1.
If we knew the values of B,/, (1) and Bt (1)) we would be able to calculate
Var(B,). So we take partial derivatives of G(w, z) with respect to z:
&G(w,z) = LB:,(z)?
7x30
= m(&z(:) %)me’ 5 (i)Z(‘)Pl $;
, /
& w,z) = xB;(z)y
3x30