Identifier
Values
['A',1] => ([],1) => [2] => [1,1] => 1
['A',2] => ([(0,2),(1,2)],3) => [3,2] => [2,2,1] => 2
['B',2] => ([(0,3),(1,3),(3,2)],4) => [4,2] => [2,2,1,1] => 2
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => [2,2,1,1,1,1] => 2
['B',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => [6,6,6,2] => [4,4,3,3,3,3] => 3
['C',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => [6,6,6,2] => [4,4,3,3,3,3] => 3
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Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
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.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
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.