PART::PartOrderMat --
order matrix on partitions
Call(s)
PART::PartOrderMat(n <,order>)
Parametersn | - | an integer |
Optionsorder=natural,lexic,cixel | - |
IntroductionThe PART::PartOrderMat function gives the order matrix on all
partitions of ListPart(n).
The considered ordering is given by the second argument kind:
natural:
part1 >= part2 if
sum_part1<=sum_part2
for the lexicographic ordering
(sum_I=[I[1], I[1]+I[2], I[1]+I[2]+I[3], ...]).
lexic:
part1 >= part2
for the lexicographic ordering.
cixel:
part1 >= part2
for the inverse lexicographic ordering.
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 FunctionsMuPAD Combinat, an open source algebraic combinatorics package