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## Symmetric Functions

### 7.1 Symmetric Functions in General

The theory of symmetric functions has many applications to enumerative combinatorics, as well as to such other branches of mathematics as group theory, Lie algebras, and algebraic geometry. Our aim in this chapter is to develop the basic combinatorial properties of symmetric functions; the connections with algebra will only be hinted at in Sections 7.18 and 7.24, Appendix 2, and in some exercises.

Let x = (x1, x2, . . .) be a set of indeterminates, and let n. A homogeneous symmetric function of degree n over a commutative ring R (with identity) is a formal power series

where (a) α ranges ...

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