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Identifier

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001364: Integer partitions ⟶ ℤ

Values

['A',1] => ([],1) => [1] => 1
['A',2] => ([(0,2),(1,2)],3) => [2,1] => 1
['B',2] => ([(0,3),(1,3),(3,2)],4) => [3,1] => 0
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [5,1] => 1
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => [3,2,1] => 0
['B',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => [5,3,1] => 0
['C',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => [5,3,1] => 0

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$F_{1} = q$

$F_{2} = 1 + 2\ q$

$F_{3} = 3$

Description

The number of permutations whose cube equals a fixed permutation of given cycle type.
For example, the permutation

$\pi=412365$ has cycle type

$(4,2)$ and

$234165$ is the unique permutation whose cube is

$\pi$ .

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

Greene-Kleitman invariant

Description

The Greene-Kleitman invariant of a poset.
This is the partition

$(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$ , where

$c_k$ is the maximum cardinality of a union of

$k$ chains of the poset. Equivalently, this is the conjugate of the partition

$(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$ , where

$a_k$ is the maximum cardinality of a union of

$k$ antichains of the poset.