Processing math: 100%

Identifier
Values
['A',1] => ([],1) => [2] => 1
['A',2] => ([(0,2),(1,2)],3) => [3,2] => 2
['B',2] => ([(0,3),(1,3),(3,2)],4) => [4,2] => 5
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => 37
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Description
The number of ways to refine the partition into singletons.
For example there is only one way to refine [2,2]: [2,2]>[2,1,1]>[1,1,1,1]. However, there are two ways to refine [3,2]: [3,2]>[2,2,1]>[2,1,1,1]>[1,1,1,1,1 and [3,2]>[3,1,1]>[2,1,1,1]>[1,1,1,1,1].
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.
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 α≺β if β−α is a simple root.