Identifier
Values
0 => ([(0,1)],2) => 0
1 => ([(0,1)],2) => 0
00 => ([(0,2),(2,1)],3) => 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
11 => ([(0,2),(2,1)],3) => 0
000 => ([(0,3),(2,1),(3,2)],4) => 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
111 => ([(0,3),(2,1),(3,2)],4) => 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
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Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.