---------------------------------------------------------------------------,
([rho]_q^1/2a.e^1/2ax - [rho]_q^-1/2a.e^-1/2ax)([rho]_q^1/2b.e^1/2bx - [rho]_q^-1/2b.e^-1/2bx)...([rho]_q^1/2l.e^1/2lx - [rho]_q^-1/2l.e^-1/2lx)

and the calculation in simple cases is practicable.

Thus Sylvester finds for the coefficient of x^n in

1
-----------------------
1 - x. 1 - x^2. 1 - x^3

[nu]^2 7 1 1
the expression ------ - -- - --(-)[nu] + --([rho]_3^[nu] + [rho]_3^-[nu]),
12 72 8 9

where [nu] = n + 3.

Sylvester's graphical method.

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,

o---o---o---o---o------
| | | | |
| | | | |
o---o---+---+---+------
| | | | |
| | | | |
o---+---+---+---+------
| | | | |
| | | | |
| | | | |

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. (321^3),
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

1
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