In general, partition problems present themselves which depend upon the solution of a number of simultaneous relations in integers of the form
λ1α1 + λ2α2 + λ3α3 + ... ≥ 0,
the coefficients λ being given positive or negative integers, and in some cases the generating function has been determined in a form which exhibits the fundamental solutions of the problems from which all other solutions are derivable by addition. (See MacMahon, Phil. Trans. vol. cxcii. (1899), pp. 351-401; and Trans. Camb. Phil. Soc. vol. xviii. (1899), pp. 12-34.)
The number of distributions of n objects (p1p2p3 ...) into parcels Method of symmetric functions. (m) is the coefficient of bm(p1p2p3 ...)xn in the development of the fraction
| 1 | ||
| (1 - bαx. 1 - bβx. 1 - bγx ... | ) | |
| × | (1 - bα²x². 1 - bαβx². 1 - bβ²x² ... | ) |
| × | (1 - bα³x³. 1 - bα²βx³. 1 - bαβγx³ ... | ) |
| ...... | ||
and if we write the expansion of that portion which involves products of the letters α, β, γ, ... of degree r in the form
1 + hr1 bxr + hr2 b2x2r + ... ,
we may write the development
Π r = ∞r = 1 (1 + hr1 bxr + hr2 b2x2r + ...),
and picking out the coefficient of bm xn we find