- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤ

Values

[1,0,1,0] => [2,1] => 0 => 0

[1,1,0,0] => [1,2] => 1 => 1

[1,0,1,0,1,0] => [3,2,1] => 00 => 0

[1,0,1,1,0,0] => [2,3,1] => 00 => 0

[1,1,0,0,1,0] => [3,1,2] => 00 => 0

[1,1,0,1,0,0] => [2,1,3] => 01 => 1

[1,1,1,0,0,0] => [1,2,3] => 11 => 1

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,0] => [3,2,1,4] => 001 => 1

[1,1,0,1,1,0,0,0] => [2,3,1,4] => 001 => 1

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

[1,1,1,0,0,1,0,0] => [3,1,2,4] => 001 => 1

[1,1,1,0,1,0,0,0] => [2,1,3,4] => 011 => 1

[1,1,1,1,0,0,0,0] => [1,2,3,4] => 111 => 1

[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 0000 => 0

[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 0000 => 0

[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 0000 => 0

[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 0000 => 0

[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 0000 => 0

[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 0000 => 0

[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 0000 => 0

[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 0000 => 0

[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 0000 => 0

[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 0000 => 0

[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 0000 => 0

[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 0000 => 0

[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 0000 => 0

[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0000 => 0

[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 0000 => 0

[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0000 => 0

[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0000 => 0

[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 0000 => 0

[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 0000 => 0

[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 0000 => 0

[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 0000 => 0

[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 0000 => 0

[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 0001 => 1

[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 0001 => 1

[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 0000 => 0

[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 0001 => 1

[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 0001 => 1

[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 0001 => 1

[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 0000 => 0

[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 0000 => 0

[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 0000 => 0

[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 0001 => 1

[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 0001 => 1

[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 0000 => 0

[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 0001 => 1

[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 0011 => 1

[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 0011 => 1

[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 0000 => 0

[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 0001 => 1

[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 0011 => 1

[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 0111 => 1

[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1111 => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 00000 => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 00000 => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 00000 => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => 00000 => 0

[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 00000 => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 00000 => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 00000 => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => 00000 => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => 00000 => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => 00000 => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 00000 => 0

[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => 00000 => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => 00000 => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 00000 => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 00000 => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 00000 => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 00000 => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => 00000 => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 00000 => 0

[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => 00000 => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => 00000 => 0

[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => 00000 => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => 00000 => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => 00000 => 0

[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => 00000 => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => 00000 => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 00000 => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 00000 => 0

[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 00000 => 0

[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 00000 => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => 00000 => 0

[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => 00000 => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => 00000 => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => 00000 => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 00000 => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 00000 => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 00000 => 0

[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 00000 => 0

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Description

The number of runs of ones in a binary word.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

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.