OF THE COMPOSITIONS OF NUMBERS. 
889 
Had neither of the symbols employed been such as obey the commutative law of 
alc^ebra, the whole number of combinations of the second order would have been 
simply 
(Pi+ib+ _Pj)_-. 
1\ ''■Vi''- Ih'- 
Thus, in this case, the symbol + introduces a complexity into the theory of 
compositions. 
Choosing similarly from h different symbols, none of which are commutative, 
there are 
(Pl_+P2±^)J , 
Pi! P2! P3! 
combinations of order h, and the general generating function may clearly be written 
1 1 
— • - 5 
Ju 1 — Zl (^1 + • ■ • + «)i) 
but there is a lesser number of combinations if one or more of the symbols be 
commutative. 
71. For the purpose of this investigation the most interesting case to consider is 
that in which one symbol, viz., the blank space, is non-comniutative, and the 
remaining h — 1 symbols commutative. This species of combination is not brought 
under view merely for the purpose of adding and discussing new complexity. Its 
introduction is absolutely vital to the investigation as clinching and confirming 
a conclusion which was momentarily assumed during the consideration of the 
compositions of multipartite numbers. (See ante, Art. 61.) Consider the reticu¬ 
lation of a multipartite number. That of a bipartite number will suffice as indicative 
of the general case. 
A combination of order h (of the nature under view) is regarded as having m parts 
when m — 1 blank space symbols occur in the combination. 
Let us first enquire how many combinations possess only a single part. We 
require nodes of k different kinds, viz., a blank space node and k — 1 other different 
nodes. A blank space node may be either essential or non-essential, but an essential 
MDCCCXCIII,—A. 5 X 
