THE PARTITION OF NUMBERS. 
85 
the enumerating generating function is the symmetric function 
_]_ _1__ 
(l—a) (l — a.a) (l — /3a) (l — ya) ... 
__ 1 _ 
(1— a 2 a) (1 —f3 2 a) (l —y 2 a) ... (l—afta) (l—a ya) (l—/3ya) ... 
x 1 _ 
(l— (fa) ... (l— a 2 /3a) ... (l — a(3ya) ... 
x. 
wherein if h, denote the sum of homogeneous products of weight s of the quantities 
a, /3, y, ..., the s — 1 th fractional factor of the generating function possesses a 
denominator factor corresponding to every separate term of h s . The function is to 
be developed in ascending powers of a and, replacing for the moment the series 
a, ,8, y, ... by a ls a 2 , a 3 , ... , we seek the coefficient of 
a k (Sa^a^ ... a/"'). 
We write this, usually, in the notation 
a k {m x m 2 ... m s ). 
The coefficient mentioned enumerates the partitions of the multipartite number 
(m 1 m 2 ...m J ), 
into k or fewer parts. If the first fractional factor l/l— a had been omitted the 
coefficient would have denoted the number of the partitions into exactly h parts. 
The inclusion of the factor l/l — a is of great importance to the investigation and 
equally yields the enumerations into exactly k parts, because from the coefficients of 
a k (m 1 m 2 ... rn s ) we have merely to subtract the coefficients of a k ~ 1 (m l m 2 ... m 3 ). The 
importance is due to the circumstance that the symmetric functions which present 
themselves in the expansion are in the best possible form for the performance of the 
Hammond operators. This is not the case when the factor l/l— a is excluded, as 
then a transformation, the necessity for which is not at once clear, is needed to obtain 
the proper form. 
I will remind the reader that, writing 
(1 — ax) (l — fix) (l — yx )... = 1 —a Y x-\-a^?—a^c z + ..., 
Hammond’s differential operator of order m is 
= —r(3 ai +CT ] 3 as + a a 3 fls + l 
m ! 
