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PART::PartOrderMat -- order matrix on partitions

Call(s)


PART::PartOrderMat(n <,order>)

Parameters

n- an integer

Options

order=natural,lexic,cixel-

Introduction

The PART::PartOrderMat function gives the order matrix on all partitions of ListPart(n).

The considered ordering is given by the second argument kind:

Without a third argument, the comparison is assumed to be natural.

The inverse matrix for this order is known as the Moebius matrix.

Example 1

>> muEC::PART::PartOrderMat( 6 );
                   +-                                 -+
                   |  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0  |
                   |                                   |
                   |  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1  |
                   +-                                 -+
>> Dom::Matrix()( % )^(-1);
              +-                                           -+
              |   1,  0,  0,  0,  0,  0,  0,  0,  0,  0, 0  |
              |                                             |
              |  -1,  1,  0,  0,  0,  0,  0,  0,  0,  0, 0  |
              |                                             |
              |   0, -1,  1,  0,  0,  0,  0,  0,  0,  0, 0  |
              |                                             |
              |   0,  0, -1,  1,  0,  0,  0,  0,  0,  0, 0  |
              |                                             |
              |   0,  0, -1,  0,  1,  0,  0,  0,  0,  0, 0  |
              |                                             |
              |   0,  0,  1, -1, -1,  1,  0,  0,  0,  0, 0  |
              |                                             |
              |   0,  0,  0,  0,  0, -1,  1,  0,  0,  0, 0  |
              |                                             |
              |   0,  0,  0,  0,  0, -1,  0,  1,  0,  0, 0  |
              |                                             |
              |   0,  0,  0,  0,  0,  1, -1, -1,  1,  0, 0  |
              |                                             |
              |   0,  0,  0,  0,  0,  0,  0,  0, -1,  1, 0  |
              |                                             |
              |   0,  0,  0,  0,  0,  0,  0,  0,  0, -1, 1  |
              +-                                           -+
>> muEC::PART::PartOrderMat( 4, cixel );
                            +-               -+
                            |  1, 1, 1, 1, 1  |
                            |                 |
                            |  0, 1, 0, 1, 1  |
                            |                 |
                            |  0, 1, 1, 1, 1  |
                            |                 |
                            |  0, 0, 0, 1, 1  |
                            |                 |
                            |  0, 0, 0, 0, 1  |
                            +-               -+

Related Functions

CompPart, ListPart

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