2^[sigma]2 x_s3^[sigma]3 ... + ...

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.

Case II.

2. _The Compositions of Multipartite Numbers. Parcels denoted by
(I^m)._--There are here no similarities between the parcels.

Let ([pi]1 [pi]2 [pi]3) be a partition of m.

(p1^[pi]1 p2^[pi]2 p3^[pi]3) 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 [pi]s parcels each contain ps objects, where s = 1, 2, 3, ...
Consider the product h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ... which can be
permuted in m! / ([pi]1![pi]2![pi]3! ...) ways. For each of these ways
h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ... will be a distribution function for
distributions of the specified type. Hence, regarding all the
permutations, the distribution function is

m!
------------------------ h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ...
[pi]1! [pi]2! [pi]3! ...

and regarding, as well, all the partitions of n into exactly m parts,
the desired distribution function is

m!
[Sigma] ------------------------ h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ...
[pi]1! [pi]2! [pi]3! ...
[ [Sigma]_[pi] = ([Sigma]_[pi])p = n ],

that is, it is the coefficient of x^n in (h1x + h2x^2 + h3x^3 + ... )^m.
The value of A_{(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...), (1^m)} is the
coefficient of (p1^[pi]1 p2^[pi]2 p3^[pi]3 ...)x^n in the development of
the above expression, and is easily shown to have the value

/p1 + m - 1\^[pi]1 /p2 + m - 1\^[pi]2 /p3 + m - 1\^[pi]3
\ p1 / \ p2 / \ p3 / ...

- /m\ /p1 + m - 2\^[pi]1 /p2 + m - 2\^[pi]2 /p3 + m - 2\^[pi]3
\1/ \ p1 / \ p2 / \ p3 / ...

- /m\ /p1 + m - 3\^[pi]1 /p2 + m - 3\^[pi]2 /p3 + m - 3\^[pi]3
\1/ \ p1 / \ p2 / \ p3 / ...

- ... to m terms.

Observe that when p1 = p2 = p3 = ... = [pi]1 = [pi]1 = [pi]1 ... = 1
this expression reduces to the mth divided differences of 0^n. The
expression gives the compositions of the multipartite number
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