[ 81 ] 
IV. Seventh Memoir on the Partition of Numbers. 
A Detailed Study of the Enumeration of the Partitions of Multipartite Numbers. 
By Major P. A. MacMahon, R.A., D.Sc., Sc.D., F.R.S. 
Received April 13,—Read May 11, 1916. 
Introduction. 
In this paper I give the complete solution of the problem of the partition of 
multipartite numbers. This is the same subject as that named by Sylvester, 
“ Compound Denumeration.” Twenty-nine years have elapsed since I announced 
that the algebra of symmetric functions is co-extensive with the grand problems of 
the combinatory analysis. The theory of symmetric function supplies generating 
functions which enumerate all the combinations, while the operators of Hammond* 
are the instruments which are effective in actually evaluating the coefficients of the 
terms of the expanded generating functions. When these operators fell from the 
hands of Hammond, they were already of much service as mining tools in extracting 
the ore from the mine field of symmetric functions ; but they were only partially 
adequate. They required sharpening and general adaptation to the work in hand. 
The first step was to decompose an operator of given order into the sum of a number 
of operators in correspondence with every partition of the number which defines the 
order. Since there is a Hammond operator corresponding to every positive integer, 
this process resulted in there being an operator in correspondence with every partition 
of every integer. The outcome of this decomposition was that the operators were 
able to deal with the symmetric operands in a much more effective manner. The 
surface material of the mine could not only be removed, but the strata to a considerable 
depth could be dealt with. But this was not sufficient. It became necessary to effect 
a further decomposition by showing that every partition operator could be represented 
by a sum of composition operators. There emerged a composition operator in 
correspondence with every permutation of the parts of the partition of the operator. 
The operators at once became effective in dealing with the material in the lower strata 
of the mine field. The operators had, in fact, been handled with particular reference 
to the operands with which they were to be associated. It was now necessary to 
deal with the material of the mine with particular reference to the tools which had 
* ‘Proc, Lond, Math, Soc,,’ 1883, vol. xiv., pp. 119-129. 
VOL. CCXVII.-A 552, N [Published June 23,1917, 
