Identifier
Values
0 => ([(0,1)],2) => 1
1 => ([(0,1)],2) => 1
00 => ([(0,2),(2,1)],3) => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
11 => ([(0,2),(2,1)],3) => 1
000 => ([(0,3),(2,1),(3,2)],4) => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
111 => ([(0,3),(2,1),(3,2)],4) => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
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Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
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 $v$.