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] => 2
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => 2
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Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
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.