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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