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Statistic identifier: St001706
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Collection: Graphs
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Description: The number of closed sets in a graph.
A subset $S$ of the set of vertices is a closed set, if for any pair of distinct elements of $S$ the intersection of the corresponding neighbourhoods is a subset of $S$:
$$
\forall a, b\in S: N(a)\cap N(b) \subseteq S.
$$
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References: [1] Koh, K. M., Poh, K. S. On the spectrum of the closed-set lattice of a graph [[MathSciNet:1165806]]
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Code:
def is_closed_set(G, S):
for u, v in Subsets(S, 2):
if not set(G[u]).intersection(G[v]).issubset(S):
return False
return True
def statistic(G):
return sum(1 for S in Subsets(G.vertices()) if is_closed_set(G, S))
-----------------------------------------------------------------------------
Statistic values:
([],0) => 1
([],1) => 2
([],2) => 4
([(0,1)],2) => 4
([],3) => 8
([(1,2)],3) => 8
([(0,2),(1,2)],3) => 7
([(0,1),(0,2),(1,2)],3) => 5
([],4) => 16
([(2,3)],4) => 16
([(1,3),(2,3)],4) => 14
([(0,3),(1,3),(2,3)],4) => 12
([(0,3),(1,2)],4) => 16
([(0,3),(1,2),(2,3)],4) => 12
([(1,2),(1,3),(2,3)],4) => 10
([(0,3),(1,2),(1,3),(2,3)],4) => 8
([(0,2),(0,3),(1,2),(1,3)],4) => 10
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
([],5) => 32
([(3,4)],5) => 32
([(2,4),(3,4)],5) => 28
([(1,4),(2,4),(3,4)],5) => 24
([(0,4),(1,4),(2,4),(3,4)],5) => 21
([(1,4),(2,3)],5) => 32
([(1,4),(2,3),(3,4)],5) => 24
([(0,1),(2,4),(3,4)],5) => 28
([(2,3),(2,4),(3,4)],5) => 20
([(0,4),(1,4),(2,3),(3,4)],5) => 20
([(1,4),(2,3),(2,4),(3,4)],5) => 16
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 13
([(1,3),(1,4),(2,3),(2,4)],5) => 20
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 16
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 13
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 9
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 13
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
([(0,4),(1,3),(2,3),(2,4)],5) => 21
([(0,1),(2,3),(2,4),(3,4)],5) => 20
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 14
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 17
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 10
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 9
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 9
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
([],6) => 64
([(4,5)],6) => 64
([(3,5),(4,5)],6) => 56
([(2,5),(3,5),(4,5)],6) => 48
([(1,5),(2,5),(3,5),(4,5)],6) => 42
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 38
([(2,5),(3,4)],6) => 64
([(2,5),(3,4),(4,5)],6) => 48
([(1,2),(3,5),(4,5)],6) => 56
([(3,4),(3,5),(4,5)],6) => 40
([(1,5),(2,5),(3,4),(4,5)],6) => 40
([(0,1),(2,5),(3,5),(4,5)],6) => 48
([(2,5),(3,4),(3,5),(4,5)],6) => 32
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 34
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 26
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 22
([(2,4),(2,5),(3,4),(3,5)],6) => 40
([(0,5),(1,5),(2,4),(3,4)],6) => 49
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 32
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 35
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 26
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 33
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 26
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 21
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 26
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 26
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 20
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 16
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,5),(1,4),(2,3)],6) => 64
([(1,5),(2,4),(3,4),(3,5)],6) => 42
([(0,1),(2,5),(3,4),(4,5)],6) => 48
([(1,2),(3,4),(3,5),(4,5)],6) => 40
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 34
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 28
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 32
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 22
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 18
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 14
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 34
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 25
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 22
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 28
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 20
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 22
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 17
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 15
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 37
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 40
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 35
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 28
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 25
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 23
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 23
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 18
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 16
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 15
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 16
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 22
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 14
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 11
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 14
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 12
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 20
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 18
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 13
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 14
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 29
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 21
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 18
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 19
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 18
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 18
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 12
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 12
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 8
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 17
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 16
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 8
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 17
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 10
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 25
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 18
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 17
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 12
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 13
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 10
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 8
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 15
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 14
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 10
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 13
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 8
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 8
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 10
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 8
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 8
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
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Created: Mar 31, 2021 at 21:51 by Martin Rubey
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Last Updated: Mar 31, 2021 at 21:51 by Martin Rubey