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## `TYP::Sf` -- how symmetric functions are encoded in `SYMF`

#### Introduction

This part describes how symmetric functions are represented in `SYMF`.

Symmetric function names are compatible with Macdonald's conventions.

First of all, one may consider two types of bases: multiplicative bases and non-multiplicative ones.

The bases declared by default are the following:

`e`:
  multiplicative basis of elementary symmetric functions. The generators of such a basis are `e[]`, `e[1]`, `e[2]`, ..., and the elements of the basis are the monomials in these variables, which are denoted by `e[i1,i2,...,ik]`, the indexing vector being a partition.

`h`:
  multiplicative basis of complete homogeneous symmetric functions. The generators of such a basis are `h[]`, `h[1]`, `h[2]`, ..., and the elements of the basis are the monomials in these variables, which are denoted by `h[i1,i2,...,ik]`, the indexing vector being a partition.

`m`:
  non-multiplicative basis of monomial symmetric functions. The elements of such a basis are indexed by partitions. For instance, `m[3,3,1]` and `m[]` are elements of the m-basis.

`p`:
  multiplicative basis of power sum symmetric functions. The generators of such a basis are `p[1]`, `p[2]`, ..., and the elements of the basis are the monomials in these variables, which are denoted by `p[i1,i2,...,ik]`, the indexing vector being a partition.

`s`:
  non-multiplicative basis of Schur functions. The elements of such a basis are indexed by partitions. For instance, `s[3,3,1]` and `s[]` are elements of the s-basis.

#### Example 1

    >> muEC::TYP::Ish( h[3,1] );

                                       TRUE

    >> muEC::TYP::Iss( s[2,2,1] );

                                       TRUE

#### Related Functions

`Ise`, `Ish`, `Ism`, `Isp`, `Iss`, `IsPart`, `SfA`

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MuPAD Combinat, an open source algebraic combinatorics package