Page:EB1911 - Volume 06.djvu/778

 The constant term is

Let aundefined be a value of x which makes the fraction infinite. The residue of

is equal to the residue of

and when =, the residue vanishes, so that we have to consider

and the residue of this is, by the first lemma,

which proves the lemma.

Take F(x) = $1⁄x^{n} (1 – x^{a}) (1 – x^{b}) ... (1 – x^{l})$ = $ƒ(x)⁄x^{n}$, since the sought number is its constant term.

Let be a root of unity which makes ƒ(x) infinite when substituted for x. The function of which we have to take the residue is

We may divide the calculation up into sections by considering separately that portion of the summation which involves the primitive qth roots of unity, q being a divisor of one of the numbers a, b, ... l. Thus the qth wave is

which, putting $1⁄_{q}$ for q and = (a + b + ... + l), may be written

and the calculation in simple cases is practicable.

Thus Sylvester finds for the coefficient of xn in

the expression

where = 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 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 and having the largest part 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 $1⁄(1 – a. 1 – ax. 1 – ax^{2}. 1 – ax^{3} ... ad inf.$ in ascending powers of a; for the coefficient of a&#8202;jxn 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 $1⁄(1 – x. 1 – x^{2} ... 1 – x^{&#8202;j}$. Therefore

The coefficient of a&#8202;jxn in the expansion of

denotes the number of ways of composing n with j or fewer parts, none of which are greater than i. The expansion is known to be

It has been established by the constructive method by F. Franklin (Amer. Jour. of Math. v. 254), and shows that the generating function for the partitions in question is

which, observe, is unaltered by interchange of i and j.

Franklin has also similarly established the identity of Euler

known as the “pentagonal number theorem,” which on interpretation shows that the number of ways of partitioning n into an even number of unrepeated parts is equal to that into an uneven number, except when n has the pentagonal form (3j2 + j), j positive or negative, when the difference between the numbers of the partitions is (−)j.

To illustrate an important dissection of the graph we will consider those graphs which read the same by columns as by lines; these are called self-conjugate. Such a graph may be obviously dissected into a square, containing say 2 nodes, and into two graphs, one lateral and one subjacent, the latter being the conjugate of the former. The former graph is limited to contain not more than parts, but is subject to no other condition. Hence the number of self-conjugate partitions of n which are associated with a square of 2 nodes is clearly equal to the number of partitions of (n – 2) into or few parts, i.e. it is the coefficient of x(n− 2) in

or of xn in

and the whole generating function is

Now the graph is also composed of angles of nodes, each angle containing an uneven number of nodes; hence the partition is transformable into one containing unequal uneven numbers. In the case depicted this partition is (17, 9, 5, 1). Hence the number of the partitions based upon a square of 2 nodes is the coefficient of aundefinedxn in the product (1 + ax) (1 + ax3) (1 + ax5) ... (1 + ax2s+1) ..., and thence the coefficient of aundefined in this product is x2 / (1 – x2. 1 – x4. 1 – x6 ... 1 – x2), and we have the expansion

Again, if we restrict the part magnitude to i, the largest angle of nodes contains at most 2i – 1 nodes, and based upon a square of 2 nodes we have partitions enumerated by the coefficient of aundefinedxn in the product (1 + ax) (1 + ax3) (1 + ax5) ... (1 + ax2i−1); moreover the same number enumerates the partition of (n – 2) into or fewer parts, of which the largest part is equal to or less than i – , and is thus given by the coefficient of x(n− 2) in the expansion of