- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3] => 11

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3] => 11

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2] => 10

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,2] => 10

[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,2] => 10

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

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

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1] => 00

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,1] => 00

[1,1,0,1,1,0,0,0] => [2,4,3,1] => [2,3,1] => 00

[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1] => 00

[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,1] => 00

[1,1,1,0,1,0,0,0] => [3,4,2,1] => [3,2,1] => 00

[1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,2,1] => 00

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4] => 111

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4] => 111

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1] => 000

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,1] => 000

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [2,3,4,1] => 000

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [2,4,3,1] => 000

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [2,4,3,1] => 000

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [2,4,3,1] => 000

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [2,4,3,1] => 000

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

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

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,4,1] => 000

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,4,1] => 000

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,2,4,1] => 000

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [3,4,2,1] => 000

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [3,4,2,1] => 000

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [3,4,2,1] => 000

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [3,4,2,1] => 000

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1] => 000

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,1] => 000

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [4,3,2,1] => 000

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [4,3,2,1] => 000

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [4,3,2,1] => 000

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5] => 1111

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5] => 1111

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4] => 1110

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,5,4] => 1110

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,5,4] => 1110

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5] => 1101

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,5] => 1101

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,4,5,3] => 1100

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,4,5,3] => 1100

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,4,5,3] => 1100

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,4,3] => 1100

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,5,4,3] => 1100

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,5,4,3] => 1100

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,5,4,3] => 1100

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5] => 1011

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,5] => 1011

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4] => 1010

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,5,4] => 1010

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,5,4] => 1010

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,3,4,2,5] => 1001

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,3,4,2,5] => 1001

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,3,4,5,2] => 1000

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,3,4,5,2] => 1000

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,3,4,5,2] => 1000

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,3,5,4,2] => 1000

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,3,5,4,2] => 1000

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,3,5,4,2] => 1000

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,3,5,4,2] => 1000

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,3,2,5] => 1001

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,3,2,5] => 1001

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,4,3,5,2] => 1000

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,4,3,5,2] => 1000

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,4,3,5,2] => 1000

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,4,5,3,2] => 1000

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,4,5,3,2] => 1000

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,4,5,3,2] => 1000

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,4,5,3,2] => 1000

>>> Load all 1054 entries. <<<

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

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.