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Matching statistic: St001053
St001053: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 2 = 1 + 1
['A',2]
=> 7 = 6 + 1
['B',2]
=> 10 = 9 + 1
Description
The second positive Fuss-Catalan number of a finite Cartan type.
The positive Fuss-Catalan numbers of a finite Cartan type are given by
$$\frac{1}{|W|}\prod (d_i+mh-2) = \prod \frac{d_i+mh-2}{d_i}$$
where the products run over all degrees of homoneneous fundamenal invariants of the Weyl group of a Cartan type.
For the second Fuss-Catalan numbers see [[St000852]] and for the positive Fuss-Catalan numbers see [[St000140]].
Matching statistic: St001827
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
Description
The number of two-component spanning forests of a graph.
A '''spanning subgraph''' is a subgraph which contains all vertices of the ambient graph. A '''forest''' is a graph which contains no cycles, and has any number of connected components. A '''two-component spanning forest''' is a spanning subgraph which contains no cycles and has two connected components.
Matching statistic: St001708
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 6 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 9 - 1
Description
The number of pairs of vertices of different degree in a graph.
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