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## `SP::TableY` -- table of all Schubert polynomials

#### Call(s)

`SP::TableY(n)`

#### Parameters

`n` | - | any positive integer denoting the degree of a symmetric group

#### Introduction

The `SP::TableY` function returns the table of all Schubert polynomials indexed by codes of permutations in Sn. The polynomials are expressed in the basis of monomials.

When called with the second argument `Pe`, it returns the table of all Schubert polynomials in Sn, expressed in the dual basis of the monomial basis, i.e. the basis of products of elementary symmetric functions on an alphabet flag. For instance, `Pe[1, 2, 0, 1, 0]` stands for the product `e1(x1,x2,x3,x4)*e2(x1,x2,x3)*e1(x1)`.

#### Example 1

    >> t:=muEC::SP::TableY(3);

                              table(
                                              2
                                [2, 1, 0] = x1  x2,
                                [1, 1, 0] = x1 x2,
                                [1, 0, 0] = x1,
                                              2
                                [2, 0, 0] = x1 ,
                                [0, 1, 0] = x1 + x2,
                                [0, 0, 0] = 1
                              )

    >> t[ [2,1,0] ];

                                        2
                                      x1  x2

    >> muEC::SP::TableY(3, Pe);

                     table(
                       [2, 1, 0] = Pe[2, 1, 0],
                       [1, 1, 0] = Pe[2, 0, 0],
                       [1, 0, 0] = Pe[1, 0],
                       [2, 0, 0] = Pe[1, 1, 0] - Pe[2, 0, 0],
                       [0, 1, 0] = Pe[1, 0, 0],
                       [0, 0, 0] = Pe[0]
                     )

#### Related Functions

`TableYY`

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MuPAD Combinat, an open source algebraic combinatorics package