This extensive theorem of algebraic reciprocity includes many known theorems of symmetry in the theory of Symmetric Functions.
The whole of the theory has been extended to include symmetric functions symbolized by partitions which contain as well zero and negative parts.
2. The Compositions of Multipartite Numbers. Parcels denoted by (Im).—There are here no similarities between the parcels. Case II.
Let (π1 π2 π3) be a partition of m.
(pπ11 pπ22 pπ33 ...) a partition of n.
Of the whole number of distributions of the n objects, there will be a certain number such that n1 parcels each contain p1 objects, and in general πs parcels each contain ps objects, where s = 1, 2, 3, ... Consider the product hπ1p1 hπ2p2 hπ3p3 ... which can be permuted in m! / π1!π2!π3! ... ways. For each of these ways hπ1p1 hπ2p2 hπ3p3 ... will be a distribution function for distributions of the specified type. Hence, regarding all the permutations, the distribution function is
| m! | hπ1p1 hπ2p2 hπ3p3 ... |
| π1!π2!π3! ... |
and regarding, as well, all the partitions of n into exactly m parts, the desired distribution function is
| Σ | m! | hπ1p1 hπ2p2 hπ3p3 ... [Σπ = m, Σπ p = n], |
| π1!π2!π3! ... |
that is, it is the coefficient of xn in (h1x + h2x² + h3x³ + ... )m. The value of A (pπ11 pπ22 pπ33 ...) is the coefficient of (pπ11 pπ22 pπ33 ...)xn in the development of the above expression, and is easily shown to have the value