This theorem is of algebraic importance; for consider the simple particular case of the distribution of objects (43) into parcels (52), and represent objects and parcels by small and capital letters respectively. One distribution is shown by the scheme
| A | A | A | A | A | B | B |
| a | a | a | a | b | b | b |
wherein an object denoted by a small letter is placed in a parcel denoted by the capital letter immediately above it. We may interchange small and capital letters and derive from it a distribution of objects (52) into parcels (43); viz.:—
| A | A | A | A | B | B | B |
| a | a | a | a | a | b | b. |
The process is clearly of general application, and establishes a one-to-one correspondence between the distribution of objects (pqr ...) into parcels (p1q1r1 ...) and the distribution of objects (p1q1r1 ...) into parcels (pqr ...). It is in fact, in Case I., an intuitive observation that we may either consider an object placed in or attached to a parcel, or a parcel placed in or attached to an object. Analytically we have
Theorem.—“The coefficient of symmetric function (pqr ...) in the development of the product hp1hq1hr1 ... is equal to the coefficient of symmetric function (p1q1r1 ...) in the development of the product hphqhr ....”
The problem of Case I. may be considered when the distributions are subject to various restrictions. If the restriction be to the effect that an aggregate of similar parcels is not to contain more than one object of a kind, we have clearly to deal with the elementary symmetric functions a1, a2, a3, ... or (1), (1²), (1³), ... in lieu of the quantities h1, h2, h3, ... The distribution function has then the value ap1aq1ar1 ... or (1p1) (1q1) (1r1) ..., and by interchange of object and parcel we arrive at the well-known theorem of symmetry in symmetric functions, which states that the coefficient of symmetric function (pqr ...) in the development of the product ap1aq1ar1 ... in a series of monomial symmetric functions, is equal to the coefficient of the function (p1q1r1 ...) in the similar development of the product apaqar....
The general result of Case I. may be further analysed with important consequences.
Write
| X1 = (1)x1, X2 = (2)x2 + (1²)x1², X3 = (3)x3 + (21)x2x1 + (1³)x1³ |