***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001706 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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 $S$: $$ \forall a, b\in S: N(a)\cap N(b) \subseteq S. $$ ----------------------------------------------------------------------------- References: [1] Koh, K. M., Poh, K. S. On the spectrum of the closed-set lattice of a graph [[MathSciNet:1165806]] ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- Created: Mar 31, 2021 at 21:51 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 31, 2021 at 21:51 by Martin Rubey