Identifier
-
Mp00114:
Permutations
—connectivity set⟶
Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤ
Values
[1,2] => 1 => ([(0,1)],2) => 1
[2,1] => 0 => ([(0,1)],2) => 1
[1,2,3] => 11 => ([(0,2),(2,1)],3) => 1
[1,3,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1] => 00 => ([(0,2),(2,1)],3) => 1
[3,1,2] => 00 => ([(0,2),(2,1)],3) => 1
[3,2,1] => 00 => ([(0,2),(2,1)],3) => 1
[1,2,3,4] => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[2,3,4,1] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[2,4,1,3] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[2,4,3,1] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[3,1,4,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[3,2,4,1] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[3,4,1,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[3,4,2,1] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,2,3] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,3,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,1,3] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,3,1] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,3,1,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,3,2,1] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,3,4,5] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,1,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,3,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,5,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,5,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,3,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,4,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,5,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,5,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,1,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,2,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,5,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,5,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,1,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,1,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,2,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,4,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,4,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,2,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,3,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,5,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,5,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,1,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,3,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,5,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,5,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,1,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,2,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,5,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,5,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,1,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,1,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,2,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,2,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,3,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,3,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,2,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,2,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,3,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,3,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,4,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,4,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,3,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,1,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,1,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,2,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,4,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,4,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,1,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,1,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,2,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,2,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,3,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,4,3,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
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Description
The number of connected components of the Hasse diagram for the poset.
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
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 $v$.
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 $v$.
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