- FindStat

Identifier

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001399: 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) => 2

[2,1,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[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

[1,2,4,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[1,3,2,4] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2

[1,3,4,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[1,4,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[1,4,3,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[2,1,3,4] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[2,1,4,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2

[2,3,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 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

[3,1,2,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[3,1,4,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1

[3,2,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 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

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Description

The distinguishing number of a poset.
This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring.
See also St000469The distinguishing number of a graph., which is the same concept for graphs.

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.

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 $v$.