THEORY OF THE PARTITION OF NUMBERS. 
643 
Art. 32. We have then a one-to-one correspondence. Each bipartite partition of 
the nature considered is represented graphically by a line of route in a tripartite 
reticulation. If we please we may regard a pair of bipartite partitions as represented 
by a line of route, for from the permutation we are also led to the complementary 
partition, 
_n _ t -2 
(p - 9. - <h p - Pi - Pz’ q — <h - Pi • • •), 
in which p x . may be both zero. 
Art. 33. It has been shown that the number of lines of route which possess 
s 21 /3a bends, 
'-'32 yfi ” ’ 
S 31 ” > 
and that this number is the coefficient of 
Xoffi 1 X 3 .A X 31 s « rxPpy 
in the development of 
(a + x 2] (3 + X 31 y) p (a (3 + X,,) 7 (a (3 -f- y)’\ 
Here s 21 = 0, and the number in question becomes 
whilst the generating function becomes 
(a -f X 31 y)'‘ (« + (3 + X 32 y) 1 (a + (3 y) r - 
In this the coefficient of 
Xo.A X 31 %1 a p /8y 
is equal to the coefficient of the same term in the expansion of the fractioi 
_ 1 __ 
1 — « — (3 — 7+«/3+(l — A S2 ) (3 7 + (1 — x 31 ) «7 — (I — A. 33 ) a/3y ' 
which is 
(1 — «) (1 — A) (1 - 7) - y «7 - X 32 £7 (1 — «) ’ 
and the verification is readily carried out. 
* ‘ Phil. Trans.,’ vol. 1S4 ( loc. cit.), Arts. 34, et seq. 
4 N 2 
