Plethystic substitution

Plethystic substitution is a shorthand notation for a common kind of substitution in the algebra of symmetric functions and that of symmetric polynomials. It is essentially basic substitution of variables, but allows for a change in the number of variables used.
Definition.
The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions formula_1 is generated as an "R"-algebra by the power sum symmetric functions
formula_2
For any symmetric function formula_3 and any formal sum of monomials formula_4, the "plethystic substitution" f[A] is the formal series obtained by making the substitutions
formula_5
in the decomposition of formula_3 as a polynomial in the "p"k's.
Examples.
If formula_7 denotes the formal sum formula_8, then formula_9.
One can write formula_10 to denote the formal sum formula_11, and so the plethystic substitution formula_12 is simply the result of setting formula_13 for each i. That is,
formula_14.
Plethystic substitution can also be used to change the number of variables: if formula_15, then formula_16 is the corresponding symmetric function in the ring formula_17 of symmetric functions in "n" variables.
Several other common substitutions are listed below. In all of the following examples, formula_8 and formula_19 are formal sums.
formula_22
formula_25, 
where formula_26 is the well-known involution on symmetric functions that sends a Schur function formula_27 to the conjugate Schur function formula_28.