THEORY OF THE PARTITION OF NUMBERS. 
621 
numbers is termed a partition ot' the original number; and this number, qua 
partitions, is termed by Sylvester the partible number. 
A partition of a number is separated into separates by writing down a set of 
partitions, each in its own brackets, such that when all the parts of the partitions are 
assembled in a single bracket and arranged in order, the partition which is separated 
is reproduced. The constituent partitions, which are the separates, are written down 
from left to right in descending numerical order as regards the weights of the 
partitions. 
N.B. The partition ( pqr) is said to have a weight p + q + r. 
The partition separated may be termed the separable partition. 
Taking as separable partition 
(PiPzPsP*Pb)> 
two separations are 
(PiPz) iPsPi) (Pb)> 
{PiPzPs) (PiP 6 )> 
and there are many others. 
If the successive weights of the separates be 
Wy w 2> w ?s - ■ 
the separation is said to have a specification 
(w v iv 2 , iv 3 , . . .); 
the specification being denoted by a partition of the weight tv of the separable 
partition. 
The degree of a separation is the sum of the highest parts of the several separates. 
If the separation be 
{Pi • • •) jl (p >: - .) 7 ' 2 (Pt ■ • -Y 3 ■ ■ • 
the multiplicity of the separation is defined by the succession of indices 
The characteristics of a separation are 
( 1 .) The weight. 
(ii.) The separable partition. 
(iii.) The specification. 
(iv.) The degree. 
(v.) The number of separates. 
(vi. ) The multiplicity. 
