632 
MAJOR P. A. MACMAHON ON THE 
gives the permutation 
aPftP-'aF'ftPaP'ftP . . . a p ‘ + 'ft q - + \ 
From what has gone before it will be seen that every permutation of a p ftfl corresponds 
to a partition of a unipartite number into parts limited in magnitude to p and in 
number to q. Every theorem in permutations of two different letters will thus yield 
a theorem in partitions of unipartite numbers. 
The North-West partition associated with the above-written permutation is easily 
seen to be (writing the parts in ascending order as regards magnitude, viz.: in 
Sylvester’s regularised orders reversed) 
V i 
_ fe _ 
+ lh lh 
_ % 
+ lh + lh 
_.4*+A, 
Pi + lh + • • ■ 4- p s +i / 
This is a partition of the unipartite 
( hPi + c h (Pi + Ps) + c h (Pi + Pa + P3) + • • • + ( h +1 (Pi E Pa + • • • + p^ + i) 
into q { -f- q 2 + q z + , . . + q s + 1 parts, the highest part being p t + p 2 +P 3 + • • • + ps + v 
If p x be zero there will be less than q parts. If q s + 1 be zero the highest part will 
be less than p. On the other hand if p x be not zero there are exactly q parts, and 
if q s + i be not zero the highest part is p. 
Art. 19. Observe that we have a fourfold correspondence, viz., between 
(l.) The lines of route in the reticulation of the bipartite pq. 
(2.) The compositions, with positive-positive contacts, of the bipartite 
number pq. 
(3.) The permutations of the letters forming the product a p /3 q . 
(4.) The partitions of all unipartite numbers into parts limited in magnitude 
to p and in number to q. 
And also, in particular, between— 
(1.) The lines of route with s right-bends or with s left-bends. 
(2.) The compositions into s + 1 parts. 
(3.) The permutations with s, a/3 or with s, /3a contacts. 
(4.) The partitions involving s different parts. 
Art. 20. The generating function for the number of lines of route through the 
reticulation which possess 5 left-bends is 
1 + 
(?V? 
W \b 
p ? 
which is the coefficient of a p /3 q in the product 
(a -f fifty (a + ft) q ; 
(see loc. cit., Art. 24), 
