Thus Sylvester finds for the coefficient of xn in
| 1 |
| 1 - x. 1 - x². 1 - x³ |
the expression
| ν² | - | 7 | - | 1 | (-)ν + | 1 | (ρν3 + ρ-ν3), |
| 12 | 72 | 8 | 9 |
where ν = n + 3.
Sylvester, Franklin, Durfee, G. S. Ely and others have evolved a constructive theory of partitions, the object of which is the contemplation of the partitions themselves, and the evolution of their properties from a Sylvester’s graphical method. study of their inherent characters. It is concerned for the most part with the partition of a number into parts drawn from the natural series of numbers 1, 2, 3 .... Any partition, say (521) of the number 8, is represented by nodes placed in order at the points of a rectangular lattice,
when the partition is given by the enumeration of the nodes by lines. If we enumerate by columns we obtain another partition of 8, viz. (321³), which is termed the conjugate of the former. The fact or conjugacy was first pointed out by Norman Macleod Ferrers. If the original partition is one of a number n in i parts, of which the largest is j, the conjugate is one into j parts, of which the largest is i, and we obtain the theorem:— “The number of partitions of any number into [i parts | i parts or fewer,] and having the largest part [equal to j | equal or less than j,] remains the same when the numbers i and j are interchanged.”
The study of this representation on a lattice (termed by Sylvester the “graph”) yields many theorems similar to that just given, and, moreover, throws considerable light upon the expansion of algebraic series.
The theorem of reciprocity just established shows that the number of partitions of n into; parts or fewer, is the same as the number of ways of composing n with the integers 1, 2, 3, ... j. Hence we can expand 1 / (1 - a. 1 - ax. 1 - ax². 1 - ax³ ... ad inf.) in ascending powers of a; for the coefficient of ajxn in the expansion is the number of ways of composing n with j or fewer parts, and this we have seen in the coefficients of xn in the ascending expansion of 1 / (1 - x. 1 - x² ... 1 - xj). Therefore