- FindStat

Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001280

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000473

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 88%●distinct values known / distinct values provided: 80%

Values

[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 3
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 4
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1]
 => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 4
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1]
 => ? = 4
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1]
 => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [6,4,3]
 => ? = 3
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1]
 => ? = 3
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [6,4,3]
 => ? = 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2]
 => ? = 4
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1]
 => ? = 4
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [6,5,2]
 => ? = 3
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1]
 => ? = 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [6,5,3]
 => ? = 3
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1]
 => ? = 3
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [6,5,3]
 => ? = 3
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2]
 => ? = 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1]
 => ? = 4
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [6,5,2]
 => ? = 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1]
 => ? = 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1]
 => ? = 3
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => ? = 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2]
 => ? = 4
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => ? = 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1]
 => ? = 3
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => ? = 3
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,7,8,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2]
 => ? = 4
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,1]
 => ? = 4
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [4,5,6,3,7,2,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => [6,4,2,1]
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,2,8,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3]
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3,1]
 => ? = 3
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [6,4,3,2,1]
 => ? = 4
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [5,4,6,3,2,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2]
 => ? = 3
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [4,5,6,3,2,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2,1]
 => ? = 3
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,2,7,8,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3]
 => ? = 3
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,2,7,8,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2]
 => ? = 4
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2,1]
 => ? = 4
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [5,4,6,2,3,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2]
 => ? = 3
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [4,5,6,2,3,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2,1]
 => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4]
 => ? = 3
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,3,2,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1]
 => ? = 3
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4]
 => ? = 3
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,3,5,2,6,7,8,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2]
 => ? = 4
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,2,3,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4]
 => ? = 3
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1]
 => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4]
 => ? = 3
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4]
 => ? = 3

Description

The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].

Matching statistic: St000133

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [2,1] => [1,2] => [1] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => [1,2] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,1,4] => [2,3,1] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,4,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,1,4] => [3,2,1] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => [2,1,3] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => [1,3,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,3,4,2] => [1,3,2] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,4,5,1,3] => [2,4,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,5,3,1,4] => [2,3,1,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,5,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => [3,2,4,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [3,2,1,5,4] => [3,2,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => [3,4,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [3,4,1,5,2] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [3,5,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,5,1,4,2] => [3,1,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,2] => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [4,2,3,1,5] => [4,2,3,1] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [4,2,5,1,3] => [4,2,1,3] => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [4,2,1,5,3] => [4,2,1,3] => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [4,5,2,1,3] => [4,2,1,3] => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,3,2] => [4,1,3,2] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [4,1,3,5,2] => [4,1,3,2] => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [5,2,3,1,4] => [2,3,1,4] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [5,2,1,4,3] => [2,1,4,3] => 2
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [5,1,3,4,2] => [1,3,4,2] => 3
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [1,3,4,5,2] => [1,3,4,2] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,5,1,6] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,4,6,1,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [2,3,4,1,6,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,3,5,4,1,6] => [2,3,5,4,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [2,3,5,6,1,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,3,5] => [2,3,5,1,6,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [2,3,6,4,1,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [2,3,1,5,6,4] => [2,3,1,5,4] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,6,7] => [2,3,4,5,6,1,8,7] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => [2,3,4,5,7,6,1,8] => [2,3,4,5,7,6,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [2,3,4,5,7,8,1,6] => [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,4,6,8,5,7] => [2,3,4,5,7,1,8,6] => [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,5,8,6] => [2,3,4,5,8,6,1,7] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,5,6] => [2,3,4,5,8,1,7,6] => ? => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,5,6,7] => [2,3,4,5,1,7,8,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => [2,3,4,6,5,7,1,8] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,3,5,4,7,8,6] => [2,3,4,6,5,8,1,7] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,8,6,7] => [2,3,4,6,5,1,8,7] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [2,3,4,6,7,8,1,5] => [2,3,4,6,7,1,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,3,5,6,8,4,7] => [2,3,4,6,7,1,8,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,3,5,7,4,8,6] => [2,3,4,6,8,5,1,7] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,3,5,7,8,4,6] => [2,3,4,6,8,1,7,5] => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3,5,8,4,6,7] => [2,3,4,6,1,7,8,5] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,4,7,8,5] => [2,3,4,7,5,8,1,6] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,3,6,4,8,5,7] => [2,3,4,7,5,1,8,6] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,4,8,5] => [2,3,4,7,8,5,1,6] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,3,6,7,8,4,5] => [2,3,4,7,8,1,6,5] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,3,6,8,4,5,7] => [2,3,4,7,1,6,8,5] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,3,7,4,5,8,6] => [2,3,4,8,5,6,1,7] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,3,7,4,8,5,6] => [2,3,4,8,5,1,7,6] => [2,3,4,5,1,7,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,7,8,4,5,6] => [2,3,4,8,1,6,7,5] => [2,3,4,1,6,7,5] => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,4,5,6,7] => [2,3,4,1,6,7,8,5] => [2,3,4,1,6,7,5] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => [2,3,5,4,6,7,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,4,3,5,7,8,6] => [2,3,5,4,6,8,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5,8,7] => [2,3,5,4,7,6,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,4,3,6,7,8,5] => [2,3,5,4,7,8,1,6] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,2,4,3,6,8,5,7] => [2,3,5,4,7,1,8,6] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,2,4,3,7,8,5,6] => [2,3,5,4,8,1,7,6] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,5,6,7] => [2,3,5,4,1,7,8,6] => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6,8,7] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,2,4,5,3,7,8,6] => [2,3,5,6,4,8,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,2,4,5,3,8,6,7] => [2,3,5,6,4,1,8,7] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,6,3,8,7] => [2,3,5,6,7,4,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [2,3,5,6,7,8,1,4] => [2,3,5,6,7,1,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,6,8,3,7] => [2,3,5,6,7,1,8,4] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,2,4,5,7,3,8,6] => [2,3,5,6,8,4,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,2,4,5,8,3,6,7] => [2,3,5,6,1,7,8,4] => [2,3,5,6,1,7,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,4,6,3,7,8,5] => [2,3,5,7,4,8,1,6] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => [2,3,5,7,4,1,8,6] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,2,4,6,7,3,8,5] => [2,3,5,7,8,4,1,6] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,4,6,7,8,3,5] => [2,3,5,7,8,1,6,4] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,2,4,7,3,5,8,6] => [2,3,5,8,4,6,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,2,4,7,3,8,5,6] => [2,3,5,8,4,1,7,6] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,2,4,7,8,3,5,6] => [2,3,5,8,1,6,7,4] => ? => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,2,4,8,3,5,6,7] => [2,3,5,1,6,7,8,4] => [2,3,5,1,6,7,4] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,5,3,4,6,8,7] => [2,3,6,4,5,7,1,8] => ? => ? = 0

Description

The "bounce" of a permutation.

Matching statistic: St001640

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [2,1] => [1,2] => [1] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => [1,2] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,1,4] => [2,3,1] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,1,4] => [3,2,1] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,1,2] => [3,1,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,2,3] => [1,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,3,1,4,5] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,4,5,1,3] => [2,4,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,1,3,5] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,5,4,1,3] => [2,4,1,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [2,5,1,3,4] => [2,1,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => [3,2,4,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2,1,4,5] => [3,2,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => [3,4,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,4,1,2,5] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [3,5,1,2,4] => [3,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,1,2,4,5] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [4,3,5,1,2] => [4,3,1,2] => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [4,3,1,2,5] => [4,3,1,2] => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [4,5,3,1,2] => [4,3,1,2] => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [4,5,1,2,3] => [4,1,2,3] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [4,1,2,3,5] => [4,1,2,3] => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,4,3,1,2] => [4,3,1,2] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 2
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 3
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,5,1,6] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,4,6,1,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [2,3,4,1,5,6] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,3,5,4,1,6] => [2,3,5,4,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [2,3,5,6,1,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [2,3,5,1,4,6] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [2,3,6,5,1,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => [2,3,6,1,4,5] => [2,3,1,4,5] => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [2,3,1,4,5,6] => [2,3,1,4,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => [2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => [2,3,4,5,7,6,1,8] => [2,3,4,5,7,6,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [2,3,4,5,7,8,1,6] => [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,4,6,8,7,5] => [2,3,4,5,7,1,6,8] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,6,8,5] => [2,3,4,5,8,7,1,6] => [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,6,5] => [2,3,4,5,8,1,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,7,6,5] => [2,3,4,5,1,6,7,8] => [2,3,4,5,1,6,7] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => [2,3,4,6,5,7,1,8] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,3,5,4,7,8,6] => [2,3,4,6,5,8,1,7] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,8,7,6] => [2,3,4,6,5,1,7,8] => [2,3,4,6,5,1,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [2,3,4,6,7,8,1,5] => [2,3,4,6,7,1,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,3,5,6,8,7,4] => [2,3,4,6,7,1,5,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,3,5,7,6,8,4] => [2,3,4,6,8,7,1,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,3,5,7,8,6,4] => [2,3,4,6,8,1,5,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3,5,8,7,6,4] => [2,3,4,6,1,5,7,8] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,5,7,8,4] => [2,3,4,7,6,8,1,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,3,6,5,8,7,4] => [2,3,4,7,6,1,5,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,5,8,4] => [2,3,4,7,8,6,1,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,3,6,7,8,5,4] => [2,3,4,7,8,1,5,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,3,6,8,7,5,4] => [2,3,4,7,1,5,6,8] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,3,7,6,5,8,4] => [2,3,4,8,7,6,1,5] => [2,3,4,7,6,1,5] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,3,7,6,8,5,4] => [2,3,4,8,7,1,5,6] => [2,3,4,7,1,5,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,7,8,6,5,4] => [2,3,4,8,1,5,6,7] => [2,3,4,1,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,7,6,5,4] => [2,3,4,1,5,6,7,8] => [2,3,4,1,5,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => [2,3,5,4,6,7,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,4,3,5,7,8,6] => [2,3,5,4,6,8,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5,8,7] => [2,3,5,4,7,6,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,4,3,6,7,8,5] => [2,3,5,4,7,8,1,6] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,2,4,3,6,8,7,5] => [2,3,5,4,7,1,6,8] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,2,4,3,7,8,6,5] => [2,3,5,4,8,1,6,7] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,7,6,5] => [2,3,5,4,1,6,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6,8,7] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,2,4,5,3,7,8,6] => [2,3,5,6,4,8,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,2,4,5,3,8,7,6] => [2,3,5,6,4,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,6,3,8,7] => [2,3,5,6,7,4,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [2,3,5,6,7,8,1,4] => [2,3,5,6,7,1,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,6,8,7,3] => [2,3,5,6,7,1,4,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,2,4,5,7,6,8,3] => ? => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,2,4,5,8,7,6,3] => [2,3,5,6,1,4,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,4,6,5,7,8,3] => [2,3,5,7,6,8,1,4] => [2,3,5,7,6,1,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,5,8,7,3] => [2,3,5,7,6,1,4,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,2,4,6,7,5,8,3] => ? => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,4,6,7,8,5,3] => [2,3,5,7,8,1,4,6] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,2,4,7,6,5,8,3] => [2,3,5,8,7,6,1,4] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,2,4,7,6,8,5,3] => [2,3,5,8,7,1,4,6] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,2,4,7,8,6,5,3] => [2,3,5,8,1,4,6,7] => ? => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,2,4,8,7,6,5,3] => [2,3,5,1,4,6,7,8] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,5,4,3,6,8,7] => [2,3,6,5,4,7,1,8] => ? => ? = 0

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

Matching statistic: St000991
(load all 2 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [2,1] => [1,2] => [1] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1,3] => [2,1] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,1,4] => [2,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [2,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,3,4] => [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,1,4] => [3,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,4,1,2] => [3,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1,2,4] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,1,2] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,3,1,4,5] => [2,3,1,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,4,5,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,1,3,5] => [2,4,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,5,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,5,3,4] => [2,1,3,4] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => [3,2,4,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2,1,4,5] => [3,2,1,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => [3,4,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,4,1,2,5] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,5,2,4] => [3,1,2,4] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,1,2,4,5] => [3,1,2,4] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [4,3,5,1,2] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [4,3,1,2,5] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [1,4,5,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [1,5,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [1,5,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,5,1,6] => [2,3,4,5,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,4,6,1,5] => [2,3,4,1,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [2,3,4,1,5,6] => [2,3,4,1,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,3,5,4,1,6] => [2,3,5,4,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [2,3,5,6,1,4] => [2,3,5,1,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [2,3,5,1,4,6] => [2,3,5,1,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [2,3,6,5,1,4] => [2,3,5,1,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => [2,3,1,6,4,5] => [2,3,1,4,5] => 3 = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [2,3,1,4,5,6] => [2,3,1,4,5] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,7] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => [2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => [2,3,4,5,7,6,1,8] => [2,3,4,5,7,6,1] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [2,3,4,5,7,8,1,6] => [2,3,4,5,7,1,6] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,4,6,8,7,5] => [2,3,4,5,7,1,6,8] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,6,8,5] => [2,3,4,5,8,7,1,6] => [2,3,4,5,7,1,6] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,8,6,7,5] => [2,3,4,5,1,8,6,7] => [2,3,4,5,1,6,7] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,7,6,5] => [2,3,4,5,1,6,7,8] => [2,3,4,5,1,6,7] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => [2,3,4,6,5,7,1,8] => ? => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,3,5,4,7,8,6] => [2,3,4,6,5,8,1,7] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,8,7,6] => [2,3,4,6,5,1,7,8] => [2,3,4,6,5,1,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [2,3,4,6,7,8,1,5] => [2,3,4,6,7,1,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,3,5,6,8,7,4] => [2,3,4,6,7,1,5,8] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,3,5,7,6,8,4] => [2,3,4,6,8,7,1,5] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,3,5,8,6,7,4] => [2,3,4,6,1,8,5,7] => ? => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3,5,8,7,6,4] => [2,3,4,6,1,5,7,8] => ? => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,5,7,8,4] => [2,3,4,7,6,8,1,5] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,3,6,5,8,7,4] => [2,3,4,7,6,1,5,8] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,7,5,6,8,4] => [2,3,4,8,6,7,1,5] => [2,3,4,6,7,1,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,3,8,5,6,7,4] => [2,3,4,1,7,8,5,6] => [2,3,4,1,7,5,6] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,3,8,5,7,6,4] => [2,3,4,1,7,5,6,8] => [2,3,4,1,7,5,6] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,3,7,6,5,8,4] => [2,3,4,8,7,6,1,5] => [2,3,4,7,6,1,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,3,8,6,5,7,4] => [2,3,4,1,8,7,5,6] => [2,3,4,1,7,5,6] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,8,6,7,5,4] => [2,3,4,1,8,5,6,7] => [2,3,4,1,5,6,7] => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,7,6,5,4] => [2,3,4,1,5,6,7,8] => [2,3,4,1,5,6,7] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => [2,3,5,4,6,7,1,8] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,4,3,5,7,8,6] => [2,3,5,4,6,8,1,7] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5,8,7] => [2,3,5,4,7,6,1,8] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,4,3,6,7,8,5] => [2,3,5,4,7,8,1,6] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,2,4,3,6,8,7,5] => [2,3,5,4,7,1,6,8] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,2,4,3,8,6,7,5] => [2,3,5,4,1,8,6,7] => ? => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,7,6,5] => [2,3,5,4,1,6,7,8] => ? => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6,8,7] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,2,4,5,3,7,8,6] => [2,3,5,6,4,8,1,7] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,2,4,5,3,8,7,6] => [2,3,5,6,4,1,7,8] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,6,3,8,7] => [2,3,5,6,7,4,1,8] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [2,3,5,6,7,8,1,4] => [2,3,5,6,7,1,4] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,6,8,7,3] => [2,3,5,6,7,1,4,8] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,2,4,5,7,6,8,3] => ? => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,2,4,5,8,7,6,3] => [2,3,5,6,1,4,7,8] => ? => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,4,6,5,7,8,3] => [2,3,5,7,6,8,1,4] => [2,3,5,7,6,1,4] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,5,8,7,3] => [2,3,5,7,6,1,4,8] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,2,4,7,5,6,8,3] => [2,3,5,8,6,7,1,4] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,4,8,5,6,7,3] => [2,3,5,1,7,8,4,6] => ? => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,2,4,7,6,5,8,3] => [2,3,5,8,7,6,1,4] => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,2,4,8,6,5,7,3] => [2,3,5,1,8,7,4,6] => ? => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,2,4,8,6,7,5,3] => ? => ? => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,2,4,8,7,6,5,3] => [2,3,5,1,4,6,7,8] => ? => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,5,4,3,6,8,7] => [2,3,6,5,4,7,1,8] => ? => ? = 0 + 1

Description

The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].

Matching statistic: St001499

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [7,5,4,3,2,1,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,7,1,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [7,5,4,3,2,6,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,7,8,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => [7,6,4,3,2,1,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [7,6,4,3,2,5,8,1] => ?
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,1,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,7,2,1,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [7,5,4,3,6,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,1,7,8] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,6,4,3,5,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,7,1,8] => ?
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,5,4,3,6,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [7,4,3,2,5,6,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,7,8,1] => ?
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [7,6,5,3,2,1,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [7,6,5,3,2,4,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => [7,5,3,2,1,4,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [7,6,5,3,4,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => [6,5,3,2,4,1,7,8] => ?
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [6,5,3,2,4,7,8,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [7,6,4,3,2,5,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [7,6,4,3,5,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [7,5,4,3,6,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,3,2,1,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [7,5,4,6,3,2,8,1] => ?
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,2,1,8] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,6,4,5,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [6,4,3,5,2,7,1,8] => ?
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,5,4,6,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,3,8,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [7,4,3,5,2,6,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [6,4,3,5,2,7,8,1] => ?
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,2,7,8,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,1,8] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => [7,6,3,2,1,4,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1

Description

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.

Matching statistic: St000989

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [2,1] => [1,2] => [1] => ? = 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => [1,2] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,1,4] => [2,3,1] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,1,4] => [3,2,1] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,1,2] => [3,1,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,2,3] => [1,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,3,1,4,5] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,4,5,1,3] => [2,4,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,1,3,5] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,5,4,1,3] => [2,4,1,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [2,5,1,3,4] => [2,1,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => [3,2,4,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2,1,4,5] => [3,2,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => [3,4,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,4,1,2,5] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [3,5,1,2,4] => [3,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,1,2,4,5] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [4,3,5,1,2] => [4,3,1,2] => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [4,3,1,2,5] => [4,3,1,2] => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [4,5,3,1,2] => [4,3,1,2] => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [4,5,1,2,3] => [4,1,2,3] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [4,1,2,3,5] => [4,1,2,3] => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,4,3,1,2] => [4,3,1,2] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 2
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 3
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,5,1,6] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,4,6,1,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [2,3,4,1,5,6] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,3,5,4,1,6] => [2,3,5,4,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [2,3,5,6,1,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [2,3,5,1,4,6] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [2,3,6,5,1,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => [2,3,6,1,4,5] => [2,3,1,4,5] => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [2,3,1,4,5,6] => [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [2,4,3,5,1,6] => [2,4,3,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => [2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => [2,3,4,5,7,6,1,8] => [2,3,4,5,7,6,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [2,3,4,5,7,8,1,6] => [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,4,6,8,7,5] => [2,3,4,5,7,1,6,8] => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,6,8,5] => [2,3,4,5,8,7,1,6] => [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,6,5] => [2,3,4,5,8,1,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,7,6,5] => [2,3,4,5,1,6,7,8] => [2,3,4,5,1,6,7] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => [2,3,4,6,5,7,1,8] => ? => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,3,5,4,7,8,6] => [2,3,4,6,5,8,1,7] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,8,7,6] => [2,3,4,6,5,1,7,8] => [2,3,4,6,5,1,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [2,3,4,6,7,8,1,5] => [2,3,4,6,7,1,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,3,5,6,8,7,4] => [2,3,4,6,7,1,5,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,3,5,7,6,8,4] => [2,3,4,6,8,7,1,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,3,5,7,8,6,4] => [2,3,4,6,8,1,5,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3,5,8,7,6,4] => [2,3,4,6,1,5,7,8] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,5,7,8,4] => [2,3,4,7,6,8,1,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,3,6,5,8,7,4] => [2,3,4,7,6,1,5,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,5,8,4] => [2,3,4,7,8,6,1,5] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,3,6,7,8,5,4] => [2,3,4,7,8,1,5,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,3,6,8,7,5,4] => [2,3,4,7,1,5,6,8] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,3,7,6,5,8,4] => [2,3,4,8,7,6,1,5] => [2,3,4,7,6,1,5] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,3,7,6,8,5,4] => [2,3,4,8,7,1,5,6] => [2,3,4,7,1,5,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,7,8,6,5,4] => [2,3,4,8,1,5,6,7] => [2,3,4,1,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,7,6,5,4] => [2,3,4,1,5,6,7,8] => [2,3,4,1,5,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => [2,3,5,4,6,7,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,4,3,5,7,8,6] => [2,3,5,4,6,8,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5,8,7] => [2,3,5,4,7,6,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,4,3,6,7,8,5] => [2,3,5,4,7,8,1,6] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,2,4,3,6,8,7,5] => [2,3,5,4,7,1,6,8] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,2,4,3,7,8,6,5] => [2,3,5,4,8,1,6,7] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,7,6,5] => [2,3,5,4,1,6,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6,8,7] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,2,4,5,3,7,8,6] => [2,3,5,6,4,8,1,7] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,2,4,5,3,8,7,6] => [2,3,5,6,4,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,6,3,8,7] => [2,3,5,6,7,4,1,8] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [2,3,5,6,7,8,1,4] => [2,3,5,6,7,1,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,6,8,7,3] => [2,3,5,6,7,1,4,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,2,4,5,7,6,8,3] => ? => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,2,4,5,8,7,6,3] => [2,3,5,6,1,4,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,4,6,5,7,8,3] => [2,3,5,7,6,8,1,4] => [2,3,5,7,6,1,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,5,8,7,3] => [2,3,5,7,6,1,4,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,2,4,6,7,5,8,3] => ? => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,4,6,7,8,5,3] => [2,3,5,7,8,1,4,6] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,2,4,7,6,5,8,3] => [2,3,5,8,7,6,1,4] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,2,4,7,6,8,5,3] => [2,3,5,8,7,1,4,6] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,2,4,7,8,6,5,3] => [2,3,5,8,1,4,6,7] => ? => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,2,4,8,7,6,5,3] => [2,3,5,1,4,6,7,8] => ? => ? = 3

Description

The number of final rises of a permutation. For a permutation $\pi$ of length $n$, this is the maximal $k$ such that $$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$ Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].

Matching statistic: St000836

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [2,1] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 3
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,2,4,5,6] => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,2,5,6] => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,3,2,1,4,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,1,6,7] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,2,4,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [6,3,2,1,4,5,7] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => ? = 1

Description

The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.

Matching statistic: St000837
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,4,6,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,5,6,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,5,3,6,2,1] => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,2,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,5,6,7,2,1] => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,4,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,2,4,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [5,4,6,3,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,3,2,7,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1

Description

The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.

Matching statistic: St000454

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001557The number of inversions of the second entry of a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000292The number of ascents of a binary word. St000877The depth of the binary word interpreted as a path. St000291The number of descents of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000390The number of runs of ones in a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.