Σ hτ1hτ2hτ3 ... ,
t1 t2 t3
where
Σ τ = m, Σ rt = n.
The quantities h are symmetric functions of the quantities α, β, γ, ... which in simple cases can be calculated without difficulty, and then the distribution function can be formed.
Ex. Gr.—Required the enumeration of the partitions of all multipartite numbers (p1p2p3 ...) into exactly two parts. We find
| h2² = h4 - h3h1 + h²2 h3² = h6 - h5h1 + h4h2 h4² = h8 - h7h1 + h6h2 + h5h3 + h²4, |
and paying attention to the fact that in the expression of hr2 the term h²r is absent when r is uneven, the law is clear. The generating function is
h2x2 + h2h1x3 + (h4 + h22)x4 + (h4h1 + h3h2)x5 + (h6 + 2h4h2)x6 + (h6h1 + h5h2 + h4h3)x7 + (h8 + 2h6h2 + h24)x8 + ...
Taking
h4 + h22 = h4 + {(2) + (12)}2 = 2(4) + 3(31) + 4(22) + 5(212) + 7(14),