PART::PartOrderMat
--
order matrix on partitions
PART::PartOrderMat(n <,order>)
n | - | an integer |
order=natural,lexic,cixel | - |
The 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.
>> 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 | +- -+
MuPAD Combinat, an open source algebraic combinatorics package