356 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OP NUMBERS, 
ttj, a,, . . . denoting the numbers of nodes in the first, second, &c., rows. 
“i ^ “2 
“2 — a* 
To find 
“i-i ^ “i 
s xr ... xr 
for all sets of integers satisfying the conditions take 
n - 
~ 1 — Xj . 1 
1 
^ Xj. 1 - 
«1 
a. 
where observe that the factors 
1 _ 1 
1 — fliXi’ 1 —(ffj/adXj’ 
. . . generate the successive rows 
of nodes and that the method of placing the letters ai, . . . ensures the satisfaction 
of the first, second, &c., conditions. 
Continued application of the simple theorem 
n 
1 
1 — ax . 1 — 
1 
1 — X . 1 — xy^ 
applied in respect of the quantities ctj, a-,, ... in succession, reduces the function 
to the form 
1 
1 - X,. 1 - XjX,. 1 - X 1 X 0 X 3 ... 1 - X,X,Xs ... X, 
the real generating function. 
The ground solutions or fundamental partitions are, as shown by the denominator 
factors, 
(oti, O62, 013, . , , Otj) 
^ (1, 0, 0, ... 0) 
( 1 , 1 , 0 , ... 0 ) 
= <; (1, 1, 1, ... 0) 
• ••••• 
Jl, 1, 1, ... 1) 
and, as might have been anticipated, the graphical representation is in evidence. 
