648 
MAJOR P. A. MACMAHON ON THE 
Art 39. We may enumerate the partitions which, excluding’ zero, involve k diffe¬ 
rent parts. Let y, 3 , s 13 , represent the number of /3y and ay contracts in a permutation. 
Id 4* >s 13 = k } the corresponding partition possesses k different parts other than 
zero. The lines of route are such as have no /3a bends, s 13 ay bends and % /3y bends. 
Pteversing the permutation we have a similar number of lines of route which have no 
a/3 bends, s 13 ya bends, and % ya bends. Now interchange a and /3 and replace the 
reticulation of the tripartite pqr by that of qpr. In this new reticulation we have 
the same number of lines of route which have no 6 a bends, s 13 y/3 bends, and % ya 
bends. This number has been shown to be 
(i) c a*) (t) L; J ■ 
Art. 40. Hence the bipartite partitions possessing k different parts other than zero 
are enumerated by 
, 'h( / fW^ + S 23 X ) / v 
k) «« \s 23/ \ s 2S / \k 
' Theorem . —Having under consideration the doubly-regularised partitions of all 
bipartite numbers into r parts, zero parts included, such that the figures of the parts 
are limited in magnitude to p and q respectively, the number of partitions which 
possess exactly k different parts, other than zero, is 
(;>! (i) j' 
s 23 assuming all compatible values. 
Th is result may be verified in the case of the tripartite 222 from the table given 
above. As an additional verification, consider the tripartite 123. For k = 2, we 
have 
Permutations. 
Compositions. 
Partitions. 
ya/3y/3y 
(bbl ITT oil) 
(bo II T2) 
yay/3 : y 
(001 101 021) 
(00 10 12) 
yay/3y/3 
(ooT To! oil) 
(bo To IT) 
a/3y‘ : /3y 
(U2 oil) 
(U 2 12) 
(111 0]2) 
(IT FP) 
(l02 02l) 
(To 2 12) 
0Ly~(3yP 
(To 2 on) 
(To 2 IT) 
ay/V/3 
(ToT 0 I 2 ) 
(To TP) 
ay/3-y" 
(To! 022 ) 
(To TP) 
