xn - a1xn-1 + a2xn-2 - ... = 0
The symmetric function Σαpβqγr..., where p + q + r + ... = n is, in the partition notation, written (pqr...). Let A(pqr...), (p1q1r1...) denote the number of ways of distributing The distribution function. the n objects defined by the partition (pqr...) into the m parcels defined by the partition (p1q1r1...). The expression
ΣA(pqr...), (p1q1r1...) · (pqr...),
where the numbers p1, q1, r1 ... are fixed and assumed to be in descending order of magnitude, the summation being for every partition (pqr...) of the number n, is defined to be the distribution function of the objects defined by (pqr...) into the parcels defined by (p1q1r1...). It gives a complete enumeration of n objects of whatever species into parcels of the given species.
1. One-to-One Distribution. Parcels m in number (i.e. m = n).—Let hs be the homogeneous product-sum of degree s of Case I. the quantities α, β, γ, ... so that
(1 - αx. 1 - βx. 1 - γx. ...)-1 = 1 + h1x + h2x² + h3x³ + ...
| h1 = Σα = (1) h2 = Σα² + Σαβ = (2) + (1²) h3 = Σα³ + Σα²β + Σαβγ = (3) + (21) + (1³). |
Form the product hp1hq1hr1...
Any term in hp1 may be regarded as derived from p1 objects distributed into p1 similar parcels, one object in each parcel, since the order of occurrence of the letters α, β, γ, ... in any term is immaterial. Moreover, every selection of p1 letters from the letters in αpβqγr ... will occur in some term of hp1, every further selection of q1 letters will occur in some term of hq1, and so on. Therefore in the product hp1hq1hr1 ... the term αpβqγr ..., and therefore also the symmetric function (pqr ...), will occur as many times as it is possible to distribute objects defined by (pqr ...) into parcels defined by (p1q1r1 ...) one object in each parcel. Hence
ΣA(pqr...), (p1q1r1...) · (pqr...) = hp1hq1hr1....