Identifier
-
Mp00148:
Finite Cartan types
—to root poset⟶
Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001488: Skew partitions ⟶ ℤ
Values
['A',1] => ([],1) => [1] => [[1],[]] => 1
['A',2] => ([(0,2),(1,2)],3) => [2,1] => [[2,1],[]] => 3
['B',2] => ([(0,3),(1,3),(3,2)],4) => [3,1] => [[3,1],[]] => 3
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Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
This is also known as the number of removable cells of the skew partition.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition (c1−c0,c2−c1,c3−c2,…), where ck is the maximum cardinality of a union of k chains of the poset. Equivalently, this is the conjugate of the partition (a1−a0,a2−a1,a3−a2,…), where ak is the maximum cardinality of a union of k antichains of the poset.
This is the partition (c1−c0,c2−c1,c3−c2,…), where ck is the maximum cardinality of a union of k chains of the poset. Equivalently, this is the conjugate of the partition (a1−a0,a2−a1,a3−a2,…), where ak is the maximum cardinality of a union of k antichains of the 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 α≺β if β−α is a simple root.
This is the poset on the set of positive roots of its root system where α≺β if β−α is a simple root.
Map
to skew partition
Description
The partition regarded as a skew partition.
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