Identifier
Values
['A',1] => ([],1) => ([],1) => 0
['A',2] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 1
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => 6
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 61
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 45
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Description
The number of proper separations of a graph.
A separation of a graph with vertex set $V$ is a set $\{A, B\}$ of subsets of $V$ such that $A\cup B = V$ and there is no edge between $A\setminus B$ and $B\setminus A$. A separation $\{A, B\}$ is proper $A\setminus B$ and $B\setminus A$ are non-empty.
For example, the number of proper separations of the empty graph on $n$ vertices $\{1,\dots,n\}$ is the Stirling number of the second kind $S(n+1, 3)$, i.e., the number of set partitions of $\{0,1,\dots,n\}$ into three subsets, where the subset containing $0$ corresponds to $A \cap B$.
Map
to graph
Description
Returns the Hasse diagram of the poset as an undirected graph.
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.