- FindStat

Your data matches 50 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000383
(load all 8 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last part of an integer composition.

Matching statistic: St000382
(load all 4 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

Matching statistic: St000297
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => 1 => 1
[1,0,1,0]
 => [1,1] => [2] => 10 => 1
[1,1,0,0]
 => [2] => [1,1] => 11 => 2
[1,0,1,0,1,0]
 => [1,1,1] => [3] => 100 => 1
[1,0,1,1,0,0]
 => [1,2] => [2,1] => 101 => 1
[1,1,0,0,1,0]
 => [2,1] => [1,2] => 110 => 2
[1,1,0,1,0,0]
 => [3] => [1,1,1] => 111 => 3
[1,1,1,0,0,0]
 => [3] => [1,1,1] => 111 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [3,1] => 1001 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => 1010 => 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [2,1,1] => 1011 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [2,1,1] => 1011 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,3] => 1100 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => 1101 => 2
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,1,2] => 1110 => 3
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => 1111 => 4
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => 1111 => 4
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,2] => 1110 => 3
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => 1111 => 4
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => 1111 => 4
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [4,1] => 10001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [3,2] => 10010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1] => 10011 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1] => 10011 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,3] => 10100 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => 10101 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [2,1,1,1] => 10111 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [2,1,1,1] => 10111 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [2,1,1,1] => 10111 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [2,1,1,1] => 10111 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,4] => 11000 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => 11001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,2,1,1] => 11011 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,2,1,1] => 11011 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => 11100 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,1,2,1] => 11101 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,1,1,2] => 11110 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,1,1,2] => 11110 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => 11111 => 5
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,10] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,10] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [1,10] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [1,10] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 10
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 9
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 10
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 9
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 10
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
 => ? => ? => ? => ? = 8
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 8
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? => ? => ? = 10
[1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 9
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => ? => ? => ? => ? = 9
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,1,0]
 => ? => ? => ? => ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
 => ? => ? => ? => ? = 10
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,8] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,8] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,9] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,10] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? => ? => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,1,8] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1

Description

The number of leading ones in a binary word.

Matching statistic: St000678

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%

Values

[1,0]
 => [1] => [1] => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 1
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,2,1,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,1,1,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,3,1,1,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,1,2,1,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,2,1,1,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,2,1,1,1,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,8] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,8] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1

Description

The number of up steps after the last double rise of a Dyck path.

Matching statistic: St000645

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. For a Dyck path

$D = D_1 \cdots D_{2n}$ with peaks in positions

$i_1 < \ldots < i_k$ and valleys in positions

$j_1 < \ldots < j_{k-1}$ , this statistic is given by

$\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)$

Matching statistic: St000439
(load all 2 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St000326

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] =>  => ? = 1
[1,0,1,0]
 => [1,2] => 1 => 1
[1,1,0,0]
 => [2,1] => 0 => 2
[1,0,1,0,1,0]
 => [1,2,3] => 11 => 1
[1,0,1,1,0,0]
 => [1,3,2] => 10 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 01 => 2
[1,1,0,1,0,0]
 => [2,3,1] => 00 => 3
[1,1,1,0,0,0]
 => [3,2,1] => 00 => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 111 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 110 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 101 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 100 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 100 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 011 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 010 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 001 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 000 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 000 => 4
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 001 => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 000 => 4
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 000 => 4
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 000 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1111 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1110 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1101 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1100 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1100 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1011 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1001 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1000 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1000 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1000 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1000 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0111 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0110 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0100 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 0100 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0011 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0010 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0001 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0000 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 0000 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 0001 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 0000 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 0000 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 0000 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0011 => 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,2,4,3,7,6,5,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,2,4,5,3,6,7,8] => ? => ? = 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] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,2,4,5,7,6,3,8] => ? => ? = 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
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,2,4,6,5,7,3,8] => ? => ? = 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] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,2,4,7,5,6,3,8] => ? => ? = 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] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,2,4,8,5,7,6,3] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,2,4,7,6,5,3,8] => ? => ? = 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] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,2,5,4,3,7,6,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,2,5,4,7,6,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,2,6,4,5,7,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,2,6,4,5,8,7,3] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,2,7,4,5,6,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,8,4,5,7,6,3] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,2,7,4,6,5,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,2,7,4,6,5,8,3] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,2,8,4,6,5,7,3] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,2,8,4,6,7,5,3] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,6,5,4,7,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,2,7,5,6,4,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,7,5,6,4,8,3] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,2,8,5,6,4,7,3] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,2,8,5,6,7,4,3] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,2,8,5,7,6,4,3] => ? => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,2,7,6,5,4,8,3] => ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,3,2,4,6,7,5,8] => ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [1,3,2,4,7,6,8,5] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,5,4,6,8,7] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,2,5,6,4,7,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,2,5,6,7,4,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,6,7,8,4] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,5,7,6,8,4] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,2,5,8,6,7,4] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,3,2,6,5,7,8,4] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,2,7,5,6,4,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [1,3,2,8,5,6,7,4] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,3,2,7,6,5,4,8] => ? => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,4,2,5,7,6,8] => ? => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,8,7,6] => ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,4,2,6,5,7,8] => ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,3,4,2,6,7,8,5] => ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,4,2,7,6,5,8] => ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,3,4,2,7,6,8,5] => ? => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,3,4,5,2,6,8,7] => ? => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,3,4,5,7,6,2,8] => ? => ? = 1

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000054
(load all 70 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation

$\pi$ of

$n$ , together with its rotations, obtained by conjugating with the long cycle

$(1,\dots,n)$ . Drawing the labels

$1$ to

$n$ in this order on a circle, and the arcs

$(i, \pi(i))$ as straight lines, the rotation of

$\pi$ is obtained by replacing each number

$i$ by

$(i\bmod n) +1$ . Then,

$\pi(1)-1$ is the number of rotations of

$\pi$ where the arc

$(1, \pi(1))$ is a deficiency. In particular, if

$O(\pi)$ is the orbit of rotations of

$\pi$ , then the number of deficiencies of

$\pi$ equals

$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$

Matching statistic: St000505
(load all 34 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%

Values

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

Description

The biggest entry in the block containing the 1.

Matching statistic: St000066
(load all 12 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%

Values

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

Description

The column of the unique '1' in the first row of the alternating sign matrix. The generating function of this statistic is given by

$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$ see [2].

The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000026The position of the first return of a Dyck path. St000025The number of initial rises of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000171The degree of the graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001725The harmonious chromatic number of a graph. St000740The last entry of a permutation. St001316The domatic number of a graph. St000501The size of the first part in the decomposition of a permutation. St001497The position of the largest weak excedence of a permutation. St000051The size of the left subtree of a binary tree. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001645The pebbling number of a connected graph. St000060The greater neighbor of the maximum. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001434The number of negative sum pairs of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001557The number of inversions of the second entry of a permutation.