t1α1, t2α2, t3α3, ... tsαs
has the expression
| ½ · | 1 | , |
| 1 - 2 (Σt1α1 - Σt1t2α1α2 + Σt1t2t3α1α2α3 - ... + (-)s + 1t1t2 ... tsα1α2 ... αs) |
and therefore the coefficient of αp11 αp22 ... αpss in the latter fraction, when t1, t2, &c., are put equal to unity, is equal to the coefficient of the same term in the product
½ (2α1 + α2 + ... + αs)p1 (2α1 + 2α2 + ... + αs)p2 ... (2α1 + 2α2 + ... + 2αs)ps.
This result gives a direct connexion between the number of compositions and the permutations of the letters in the product αp11 αp22 ... αpss. Selecting any permutation, suppose that the letter ar occurs qr times in the last pr + pr+1 + ... + ps places of the permutation; the coefficient in question may be represented by ½ Σ2q1+q2+ ... +qs, the summation being for every permutation, and since q1 = p1 this may be written
2p1-1 Σ2q1+q2+ ... +qs.
Ex. Gr.—For the bipartite 22, p1 = p2 = 2, and we have the following scheme:—
| α1 | α1 | α2 | α2 | q2 = 2 |
| α1 | α2 | α1 | α2 | = 1 |
| α1 | α2 | α2 | α1 | = 1 |
| α2 | α1 | α1 | α2 | = 1 |
| α2 | α1 | α2 | α1 | = 1 |
| α2 | α2 | α1 | α1 | = 0 |
Hence