THEORY OF THE PARTITION OF NUMBERS. 
647 
a lh y r ' a lh y r ' x . . . 
(3 tl y ri (3 q *y r ' x . . . 
which express the bipartite lilies of route which are the projections on the planes xz, 
yz respectively. Since the tripartite permutation involves no /5a contacts, we see 
that these two permutations uniquely determine the permutation 
a p ‘(3' h y r 'u. lh /3 q -y r * . . . 
Hence the number of lines of route in question is 
Art. 38. Hence also the interesting summation formula 
(l + %\ 
V S 31 / 
i r r){K r y 
Observe that the expression further enumerates the lines of route with r, (3a 
bends in the reticulation of the bipartite p -fi r > C I + r - 
A generating function which enumerates these partitions is 
_____ 1 _ 
1 — x. 1 — a . 1 — ax. . . 1 — axP 1 — y .1 — b.l — by. . .1 — byt 
in which the coefficient of ( abxi J y ri ) r must be sought. 
The compositions that appear are the principal ones along lines of route which have 
no (3a bends. We may strike out the last part of the composition whenever its last 
figure is zero, and then the compositions are not of the single tripartite 222, but of 
the 9 tripartites extending from 222 to 002, the last figure being 2, and the first two 
figures not exceeding 2, 2 respectively. The compositions are into 2, or fewer parts. 
Generally the compositions appear of the (p + 1) (</ + 1) tripartites extending from 
pqr to OOr, the last figure being r, and the first two figures not exceeding p, q, 
respectively. The compositions are into r, or fewer parts, no part having the last 
figure zero. 
The partitions present themselves in complementary pairs. To every partition 
(ab cd . . .) corresponds another (p — a, q — b, p — c, q — d . . .) the numbers par¬ 
titioned being respectively a + c fi- ..., b + d fi-... and rp — a — c — rq — b — d—... 
Ex. gr ., the complementary partitions (02 22), (20 00) of the bipartites 24, 20 
Certain partitions are self-complementary. The number partitioned is then \rp, \iq. 
