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## `PART::Part2ListHook` -- calculates the list of hook lengths

#### Call(s)

`PART::Part2ListHook(part)`

#### Parameters

`part` | - | any list denoting a partition

#### Introduction

The `PART::Part2ListHook` function computes the list of all hook lengths of a given partition. The result is sorted into non-increasing order.

The arm and leg length of a cell (i,j) are respectively part[i] - j and cpart[j] - i where `cpart` is the conjugate of `part`.

The call ``PART::Part2ListHook`(part,a)` computes the Jack-hook-product. `a` may be any expression. This is the product a*arm_length[i,j] + leg_length[i,j] + 1 over all cells of the diagram of the partition part.

The call ``PART::Part2ListHook`(part,q,t)` computes the two-variable hook polynomial. The arguments `q` and `t` can be any expressions. This is the product 1 - q^arm_length[i,j] x t^(1+leg_length[i,j]) over all cells of the diagram of the partition part.

#### Example 1

    >> muEC::PART::Part2ListHook( [3,1] );

                                   [4, 2, 1, 1]

    >> 4! / _mult( op(%) );

                                         3

    >> muEC::PART::Part2ListHook( [2,1], z );

                                       z + 2

    >> muEC::PART::Part2ListHook( [2,1], z, t );

                                   2               2
                               - (t  z - 1) (t - 1)

#### Related Functions

`Part2Conjugate`, `Part2Border`, `Part2Diagonal`, `Part2Mat`, `ListPart`, `TYP::IsPart`

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MuPAD Combinat, an open source algebraic combinatorics package