Identifier
Values
['A',1] => ([],1) => [2] => [1,1] => 1
['A',2] => ([(0,2),(1,2)],3) => [3,2] => [2,2,1] => 4
['B',2] => ([(0,3),(1,3),(3,2)],4) => [4,2] => [2,2,1,1] => 5
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => [2,2,1,1,1,1] => 7
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Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
to root poset
Description
The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.