Identifier
Values
0 => 0 => ([(0,1)],2) => 0
1 => 1 => ([(0,1)],2) => 0
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
01 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
10 => 00 => ([(0,2),(2,1)],3) => 0
11 => 11 => ([(0,2),(2,1)],3) => 0
000 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
001 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 0
010 => 000 => ([(0,3),(2,1),(3,2)],4) => 0
011 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 0
101 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
110 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
111 => 111 => ([(0,3),(2,1),(3,2)],4) => 0
1010 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
11111 => 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
flag zeros to zeros
Description
Return a binary word of the same length, such that the number of zeros equals the number of occurrences of $10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.
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 $v$.