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{{ main | Symmetric polynomial }}
The study of symmetric functions is based on that of symmetric polynomials. In a [[polynomial ring]] in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an [[group action|action]] by [[ring homomorphism|ring automorphism]]s of the [[symmetric group]] ''S<sub>n</sub>'' on the polynomial ring in ''n'' indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The [[Invariant_
: <math>X_1+X_2+\cdots+X_n, \, </math>
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=== Generating functions ===
The first definition of Λ<sub>''R''</sub> as a subring of ''R''[[''X''<sub>1</sub>,''X''<sub>2</sub>,…]] allows the [[generating function]]s of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to Λ<sub>''R''</sub>, these expressions involve operations taking place in ''R''<nowiki>[[</nowiki>''X''<sub>1</sub>,''X''<sub>2</sub>,…;''t''<nowiki>]]</nowiki> but outside its subring Λ<sub>''R''</sub>{{brackets|''t''}}, so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates ''X''<sub>''i''</sub>. We shall write "(''X'')" after the symmetric functions to stress this interpretation.
The generating function for the elementary symmetric functions is
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