Identifier
Values
['A',1] => ([],1) => ([],1) => 0
['A',2] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 3
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => 12
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 360
['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) => 360
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The normalized Knill dimension of a graph.
The Knill dimension [1], [2] is a rational number associated with a graph as follows: for the empty graph $\dim(G) = -1$. For a graph with non-empty vertex set $V$, it is $\dim(G) = 1 + \frac{1}{|V|}\sum_{v\in V}\dim(N_v)$, where
$N_v$ is the subgraph of $G$ induced by the set of neighbours of $v$.
Conjecturally, the least common multiple of the denominators of all graphs is the order of the alternating group oeis:A001710. Thus, the normalized Knill dimension is the Knill dimension multiplied with the number.
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.