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## `SYMF::SfPlethysm` -- plethysm of symmetric functions

#### Call(s)

`SYMF::SfPlethysm(sf1,sf2 <,b1> <,b2>)`

#### Parameters

`sf1, sf2` | - | any symmetric functions
`b1, b2` | - | any known bases

#### Introduction

The `SYMF::SfPlethysm` function computes the plethysm `sf1[sf2]`.

One may specify that `sf1` and `sf2` are expressed on the bases `b1` and `b2` by using either ``SYMF::SfPlethysm`(sf1, sf2, b1)`, ``SYMF::SfPlethysm`(sf1, sf2, b1, b2)`.

The plethysm operation is defined as follows: let `p[i]` be the `i`-th power-sum symmetric function and `sf1`, `sf2`, `sf3` be any symmetric functions, then,

              (sf1 + sf2)[sf3] = sf1[sf3] + sf2[sf3]
              (sf1 x sf2)[sf3] = sf1[sf3] x sf2[sf3]
              sf1[p[i]] = p[i][sf1]
              p[i][p[j]] = p[i x j]

The default is to compute plethysms on the `p`-basis and return the result expressed on the `p`-basis.

Special algorithms have been included to compute plethysms. In particular, the result is expressed on the basis of Schur functions when:

- `b1=e`, `b2=e,h,s`,
- `b1=h`, `b2=e,h,s`,
- `b1=s`, `b2=e,h,s`,
- `b1=p`, `b2=e,h,p,s,m`,
- `b1=m`, `b2=e,h,p,s,m`.

#### Example 1

    >> muEC::SYMF::SfPlethysm( p[1]*s[2], h[1] );

                               p[2, 1]   p[1, 1, 1]
                               ------- + ----------
                                  2           2

    >> muEC::SYMF::SfPlethysm( e[3], s[2,1], e, s );

          s[2, 2, 2, 1, 1, 1] + s[3, 2, 2, 1, 1] + s[3, 3, 1, 1, 1] +

             s[4, 2, 1, 1, 1] + s[3, 2, 2, 2] + s[3, 3, 2, 1] +

             s[4, 2, 2, 1] + 2 s[4, 3, 1, 1] + s[5, 2, 1, 1] +

             s[6, 1, 1, 1] + s[3, 3, 3] + s[4, 3, 2] + s[5, 2, 2] +

             s[4, 4, 1] + s[5, 3, 1]

#### Related Functions

`Char2Sf`

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MuPAD Combinat, an open source algebraic combinatorics package