___________
p1^[pi]1 p2^[pi]2 p3^[pi]3 ...

into m parts. Summing the distribution function from m = 1 to w = [oo]
and putting x = 1, as we may without detriment, we find that the
totality of the compositions is given by

h1 + h2 + h3 + ...
---------------------- which may be given the form
1 - h1 - h2 - h3 + ...

a1 - a2 + a3 - ...
-------------------------.
1 - 2(a1 - a2 + a3 - ...)

Adding 1/2 we bring this to the still more convenient form

1
1/2 -------------------------.
1 - 2(a1 - a2 + a3 - ...)

Let F(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...) denote the total number of
compositions of the multipartite /{p1^[pi]1 p2^[pi]2 p3^[pi]3}....
Then 1/2{1/1 - 2[alpha]} = 1/2 + [Sigma]F(p)[alpha]^p, and thence
F(p) = 2^(p-1).

1
Again 1/2 --------------------------------------- =
1 - 2([alpha] + [beta] - [alpha][beta])

= 1/2 + [Sigma]F(p)[alpha]^p1 [beta]^p2,

and expanding the left-hand side we easily find

(p1 + p2)! (p1 + p2 - 1)!
F(p1p2) = 2^(p1+p2-1) ---------- - 2^(p1+p2-2) ---------------------
0! p1! p2! 1!(p1 - 1)! (p2 - 1)!

(p1 + p2 - 2)!
+ 2^(p1+p2-3) --------------------- - ....
2!(p1 - 2)! (p2 - 2)!

We have found that the number of compositions of the multipartite
/(p1p2p3 ... ps) is equal to the coefficient of symmetric function
(p1p2p3...ps) _or_ of the single term [alpha]1^p1 [alpha]2^p2
[alpha]3^p3 ... [alpha]s^ps in the development according to ascending
powers of the algebraic fraction

1
1/2 . ----------------------------------------------------------------------------------------------------.
1 - 2([Sigma][a]1 - [Sigma][a]1 [a]2 + [Sigma][a]1 [a]2 [a]3) - ... + (-)^(s+1)[a]1 [a]2 [a]3...[a]s

This result can be thrown into another suggestive form, for it can be
proved that this portion of the expanded fraction

1
1/2 . -------------------------------------------------------------------------------------------------------------------,
{1 - t1(2[a]1 + [a]2 + ... + [a]3)} {1 - t2(2[a]1 + 2[a]2 + ... + [a]s)} ... {1 - t_s(2[a]1 + 2[a]2 + ... + 2[a]s)}

which is composed entirely of powers of

t1[alpha]1, t2[alpha]2, t3[alpha]3, ... t_s[alp