106 
MAJOR P. A. MacMAHON : SEVENTH MEMOIR ON 
Ex. gr. The multipartite number 3221 has the five partitions 
(1111 1110 1000), (1111 1100 1010), 
(mo mo iooi), (mo ion lioo), (mo 1101 1010 ). 
Art. 17. Again, to pass to a different restriction, if no integer constituent of a 
multipartite part is to exceed 2, we strike out from the Q, functions all partitions 
which involve integers greater than 2 and arrive at an infinite set of B functions 
which can be dealt with in a similar way. Thus 
B, = i + (i) + (2) + (1 2 ) + (21) + (1 3 ) + (2 2 ) + (21 2 ) + (1 4 ) + ... ad inf. 
B, = 1 +(2) + (4) + (2 3 ) + (42) + (2 3 ) + (4 3 ) + (42 2 ) + (2 4 )+ ... „ „ 
B, = ! + (i) + (2 i) + (i 2 ) + (2 i, i) + (r) + (2 i, 2 i) + (2 i, i , i) + (d) + ... „ „ 
In regard to the Hammond operators 
D,B, = IffB, = B,; 
while every other operator causes B, to vanish. 
To find the effect of L) m upon the product 
B/'Bff ... B/ ; 
we observe that D„, operates through the compositions of m into exactly l\ + k 2 + ... + k { 
parts, zero counting as a part. In order that a particular composition operator shall 
not cause the product to vanish, the k s factors of B/ £s must only be operated upon by 
D 0 (ee1), D s and 1)_, S . Hence the number of compositions which multiply the product 
by unity and not by zero is given by the coefficient of x m in the development of 
w 
hich is 
(1 + x + a? 2 ) 7 " (1 + x 2 + x' l ) hl ... (1 + x 1 + x 21 
' 1 — x 3 \ kl /1 — x G \ ki ( 1 — x SlX ki 
1 —X 
1 
1 — x l l 
This establishes that the effect of Iff upon the product is to multiply it by this 
coefficient. 
The generating function is 
1 + </B 1 
+ fi(B 1 3 + 3B 1 B ? +2B a ) 
o I 
+ ... 
WE,(B) 
+ ... 
