IV. Seventh memoir on the partition of numbers. A detailed study of the enumeration of the partitions of multipartite numbers

Source abstract

Abstract 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 been forged. To evaluate the coefficients we have to operate repeatedly with the appropriate operators until a numerical result is reached. In order to accomplish with facility and to establish laws we have to put the generating functions form that these operations are carried out in a regular and simple manner. To make my meaning clear, I will instance the case of the simple operation of differentiation and the exponential function eαx. We have ∂xeαx=aeαx, the effect of the operation being, to merely the operand by magnitude α.