.......
and generally
Xs = Σ(λμν ...) xλ xμ xν ...
the summation being in regard to every partition of s. Consider the result of the multiplication—
Xp1Xq1Xr1 ... = ΣP xσ1s1 xσ2s2 xσ3s3 ...
To determine the nature of the symmetric function P a few definitions are necessary.
Definition I.—Of a number n take any partition (λ1λ2λ3 ... λs) and separate it into component partitions thus:—
(λ1λ2) (λ3λ4λ5) (λ6) ...
in any manner. This may be termed a separation of the partition, the numbers occurring in the separation being identical with those which occur in the partition. In the theory of symmetric functions the separation denotes the product of symmetric functions—
Σ αλ1 βλ2 Σ αλ3 βλ4 γλ5 Σ αλ6 ...