| ( | p1 + m - 1 | ) | π1 | ( | p2 + m - 1 | ) | π2 | ( | p3 + m - 1 | ) | π3 | ... | ||||
| p1 | p2 | p3 | ||||||||||||||
| - | ( | m | ) | ( | p1 + m - 2 | ) | π1 | ( | p2 + m - 2 | ) | π2 | ( | p3 + m - 2 | ) | π3 | ... |
| 1 | p1 | p2 | p3 | |||||||||||||
| + | ( | m | ) | ( | p1 + m - 3 | ) | π1 | ( | p2 + m - 3 | ) | π2 | ( | p3 + m - 3 | ) | π3 | ... |
| 2 | p1 | p2 | p3 | |||||||||||||
| - ... to m terms. | ||||||||||||||||
Observe that when p1 = p2 = p3 = ... = π1 = π2 = π3 ... = 1 this expression reduces to the mth divided differences of 0n. The expression gives the compositions of the multipartite number pπ11 pπ22 pπ33 ... into m parts. Summing the distribution function from m = 1 to m = ∞ and putting x = 1, as we may without detriment, we find that the totality of the compositions is given by (h1 + h2 + h3 + ...) / (1 - h1 - h2 - h3 + ...) which may be given the form (a1 - a2 + a3 - ...) / [1 - 2(a1 - a2 + a3 - ...)] Adding ½ we bring this to the still more convenient form
| ½ | 1 | . |
| 1 - 2(a1 - a2 + a3 - ...) |
Let F (pπ11 pπ22 pπ33 ... ) denote the total number of compositions of the multipartite pπ11 pπ22 pπ33 .... Then ½ · (1 / 1 - 2a) = ½ + Σ F(p)αp, and thence F(p) = 2p - 1. Again ½ · [1 / 1 - 2(α + β - αβ)] = Σ F(p1p2) αp1βp2, and expanding the left-hand side we easily find
| F(p1p2) = 2p1 + p2 - 1 | (p1 + p2)! | - 2p1 + p2 - 2 | (p1 + p2 - 1)! | + 2p1 + p2 - 3 | (p1 + p2 - 2)! | - ... |
| 0! p1! p2! | 1! (p1 - 1)! (p2 - 1)! | 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 αp11 αp22 αp22 ... αpss in the development according to ascending powers of the algebraic fraction
| ½ · | 1 | . |
| 1 - 2 (Σα1 - Σα1α2 + Σα1α2α3 - ... + (-)S + 1 α1α2α3 ... αs |
This result can be thrown into another suggestive form, for it can be proved that this portion of the expanded fraction
| ½ · | 1 | , |
| {1 - t1 (2α1 + α2 + ... + αs)} {1 - t2 (2α1 + 2α2 + ... + αs)} ... {1 - ts (2α1 + 2α2 + ... + 2αs)} |
which is composed entirely of powers of