FUNCTION: PartOrderMat - order matrix on partitions

CALLING SEQUENCE:

PartOrderMat(n)

PartOrderMat(n, kind)

PART[PartOrderMat](n)

PART[PartOrderMat](n, kind)

PARAMETERS:

n = any positive integer

kind = 'natural', 'lexic', 'cixel'

SYNOPSIS:

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

• The considered order is given by the second argument kind:
  'natural' : I>=J if sum_I<=sum_J for the lexicographic ordering (sum_I=[I[1], I[1]+I[2], I[1]+I[2]+I[3], ...]).
  'lexic' : I>=J for the lexicographic ordering.
  'cixel' : I>=J for the inverse lexicographic ordering.

• The order is by default the natural order. The inverse matrix for this order is known as the Moebius matrix.

• Whenever there is a conflict between the function name PartOrderMat and another name used in the same session, use the long form PART['PartOrderMat'].

EXAMPLES:

> with(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 ]
 
> linalg[inverse](PartOrderMat(6));
 
                 [  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 ]
 
> 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 ]