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

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The biggest entry in the block containing the 1.

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

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => 1
[1,0,1,0]
 => [1,1] => 1
[1,1,0,0]
 => [2] => 2
[1,0,1,0,1,0]
 => [1,1,1] => 1
[1,0,1,1,0,0]
 => [1,2] => 1
[1,1,0,0,1,0]
 => [2,1] => 2
[1,1,0,1,0,0]
 => [3] => 3
[1,1,1,0,0,0]
 => [3] => 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1
[1,0,1,1,0,1,0,0]
 => [1,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => 2
[1,1,0,1,0,0,1,0]
 => [3,1] => 3
[1,1,0,1,0,1,0,0]
 => [4] => 4
[1,1,0,1,1,0,0,0]
 => [4] => 4
[1,1,1,0,0,0,1,0]
 => [3,1] => 3
[1,1,1,0,0,1,0,0]
 => [4] => 4
[1,1,1,0,1,0,0,0]
 => [4] => 4
[1,1,1,1,0,0,0,0]
 => [4] => 4
[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]
 => [1,1,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5] => 5
[1,1,0,1,0,1,1,0,0,0]
 => [5] => 5
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 4
[1,1,0,1,1,0,0,1,0,0]
 => [5] => 5
[1,1,0,1,1,0,1,0,0,0]
 => [5] => 5
[1,1,0,1,1,1,0,0,0,0]
 => [5] => 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,6,2] => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,2] => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2,2] => ? = 6
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2,2] => ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,6,2,2] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => ? = 8
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => ? = 8
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,6,2,2] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => ? = 8
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,2,6,2] => ? = 2
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,8,2] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,8,2] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,8,2] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,8,2] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,2,6,2] => ? = 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,8,2] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [10,2] => ? = 10
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,2,2,6] => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2,6] => ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,4,6] => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,10] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,2,8] => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [4,8] => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,10] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [2,10] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [2,10] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [12] => ? = 12
[1,1,0,0,1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [2,10] => ? = 2

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: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 91%

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
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2,2,2] => [1,2,2,2,2,1] => 1101010101 => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2,2] => [1,1,1,2,2,2,1] => 1111010101 => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2] => [1,1,1,1,1,2,2,1] => 1111110101 => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,2,2,4] => [1,2,2,2,1,1,1] => 1101010111 => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,2,6] => [1,2,2,1,1,1,1,1] => 1101011111 => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,8] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2] => [1,1,1,1,1,2,2,1] => 1111110101 => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,2,2] => [1,2,1,1,2,2,1] => 1101110101 => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,6,2] => [1,2,1,1,1,1,2,1] => 1101111101 => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,2,4,2] => [1,2,2,1,1,2,1] => 1101011101 => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2] => [1,1,1,2,1,1,2,1] => 1111011101 => ? = 4
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,2] => [1,2,1,1,1,1,2,1] => 1101111101 => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,2,6] => [1,2,2,1,1,1,1,1] => 1101011111 => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,6] => [1,1,1,2,1,1,1,1,1] => 1111011111 => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,8] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [2,8] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [2,8] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,4] => [1,1,1,2,2,1,1,1] => 1111010111 => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4] => [1,1,1,1,1,2,1,1,1] => 1111110111 => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,4] => [1,2,1,1,2,1,1,1] => 1101110111 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4] => [1,1,1,1,1,2,1,1,1] => 1111110111 => ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [2,8] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,6] => [1,1,1,2,1,1,1,1,1] => 1111011111 => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2,2,2,2] => [1,2,2,2,2,2,1] => 110101010101 => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2,2,2] => [1,1,1,2,2,2,2,1] => 111101010101 => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2,2] => [1,1,1,1,1,2,2,2,1] => 111111010101 => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,2,2,2] => [1,2,1,1,2,2,2,1] => 110111010101 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2,2] => [1,1,1,1,1,2,2,2,1] => 111111010101 => ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,6,2,2] => [1,2,1,1,1,1,2,2,1] => 110111110101 => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => [1,1,1,1,1,1,1,2,2,1] => 111111110101 => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => [1,1,1,1,1,1,1,2,2,1] => 111111110101 => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [8,2,2] => [1,1,1,1,1,1,1,2,2,1] => 111111110101 => ? = 8
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,4,2,2] => [1,2,2,1,1,2,2,1] => 110101110101 => ? = 2

Description

The number of leading ones in a binary word.

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: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 91%

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
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,1,2,4,8] => [5,6,4,7,2,3,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 8 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,1,2,6,8] => [5,6,3,4,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 8 - 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,1,2,7,8] => [5,6,1,2,3,4,7,8] => ?
 => ? = 8 - 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,1,2,6,7,8] => [4,5,1,2,3,6,7,8] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 8 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => [4,2,3,5,6,7,8,9,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 - 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [9,10,7,8,5,6,3,1,2,4] => [8,9,7,10,5,6,3,4,1,2] => ?
 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [9,10,7,8,5,3,4,1,2,6] => [8,9,6,7,5,10,3,4,1,2] => ?
 => ? = 6 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,6,3,4,1,2,8] => [8,9,6,7,4,5,3,10,1,2] => ?
 => ? = 8 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,6,3,4,1,2,10] => [8,9,6,7,4,5,2,3,1,10] => ?
 => ? = 10 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [9,7,8,10,5,6,3,4,1,2] => ? => ?
 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,6,10,3,4,1,2] => ? => ?
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,6,3,4,10,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [9,10,7,8,5,3,1,2,4,6] => [7,8,6,9,5,10,3,4,1,2] => ?
 => ? = 6 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [9,10,7,8,5,3,4,6,1,2] => ? => ?
 => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,3,4,6,8,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,3,4,6,1,2,8] => ? => ?
 => ? = 8 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,3,4,1,2,6,8] => ? => ?
 => ? = 8 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,3,1,2,4,6,8] => [6,7,5,8,4,9,3,10,1,2] => ?
 => ? = 8 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,6,8,3,4,1,2] => [9,10,7,8,4,5,3,6,1,2] => ?
 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,6,8,3,1,2,4] => [8,9,7,10,4,5,3,6,1,2] => ?
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,6,3,1,2,4,8] => [7,8,6,9,4,5,3,10,1,2] => ?
 => ? = 8 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [9,10,7,5,6,3,4,8,1,2] => [9,10,6,7,4,5,3,8,1,2] => ?
 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,8,10,3,4,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,8,10,3,1,2,4] => ? => ?
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,8,3,1,2,4,10] => ? => ?
 => ? = 10 - 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,8,3,4,10,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,8,3,4,1,2,10] => ? => ?
 => ? = 10 - 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,3,4,1,2,8,10] => ? => ?
 => ? = 10 - 1
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,3,1,2,4,8,10] => ? => ?
 => ? = 10 - 1
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,7,5,3,4,6,1,2,8,10] => ? => ?
 => ? = 10 - 1
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,7,5,3,4,1,2,6,8,10] => [6,7,4,5,3,8,2,9,1,10] => ?
 => ? = 10 - 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,3,4,8,10,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,7,5,6,3,4,8,1,2,10] => [8,9,5,6,3,4,2,7,1,10] => ?
 => ? = 10 - 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,7,5,3,4,6,8,10,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,7,5,3,4,6,8,1,2,10] => ? => ?
 => ? = 10 - 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [9,7,8,10,5,6,3,1,2,4] => ? => ?
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [9,7,8,10,5,3,1,2,4,6] => ? => ?
 => ? = 6 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [9,7,8,10,5,3,4,6,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [9,7,8,10,5,3,4,1,2,6] => ? => ?
 => ? = 6 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,3,4,6,10,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,3,4,6,1,2,10] => [8,9,5,6,4,7,2,3,1,10] => ?
 => ? = 10 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,3,4,1,2,6,10] => [7,8,5,6,4,9,2,3,1,10] => ?
 => ? = 10 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,3,1,2,4,6,10] => ? => ?
 => ? = 10 - 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,6,10,3,1,2,4] => ? => ?
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [9,7,8,5,6,3,1,2,4,10] => [7,8,6,9,4,5,2,3,1,10] => ?
 => ? = 10 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [11,12,9,10,7,8,5,6,3,4,1,2] => [11,12,9,10,7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [11,12,9,10,7,8,5,6,3,1,2,4] => ? => ?
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [11,12,9,10,7,8,5,3,1,2,4,6] => ? => ?
 => ? = 6 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [11,12,9,10,7,8,5,3,4,6,1,2] => ? => ?
 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [11,12,9,10,7,8,5,3,4,1,2,6] => ? => ?
 => ? = 6 - 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: St000326

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 91%

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
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,8,5] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ? => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ? => ? = 8
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ? => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ? => ? = 8
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [8,4,3,7,6,5,2,1] => ? => ? = 8
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7,10,9] => ? => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,5,4,6,1,8,7,10,9] => ? => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,5,4,7,6,8,1,10,9] => ? => ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,6,5,9,8,10,7] => ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,9,8,10,5] => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,9,8,10,3] => ? => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7,10,9] => ? => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7,10,9] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,1,7,5,4,6,8,3,10,9] => ? => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [3,2,7,5,4,6,8,1,10,9] => ? => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,3,2,5,4,6,8,1,10,9] => ? => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,4,3,5,2,6,8,1,10,9] => ? => ? = 8
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,4,3,7,6,8,5,10,9] => ? => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,4,1,7,6,8,5,10,9] => ? => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,7,6,8,1,10,9] => ? => ? = 8
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,5,4,7,6,8,3,10,9] => ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,4,3,9,7,6,8,10,5] => ? => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,4,1,9,7,6,8,10,5] => ? => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,2,4,9,7,6,8,10,1] => ? => ? = 10
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,5,4,9,7,6,8,10,3] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,5,4,9,7,6,8,10,1] => ? => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,3,2,5,4,7,6,8,10,1] => ? => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,4,3,5,2,7,6,8,10,1] => ? => ? = 10
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,3,2,6,5,7,4,8,10,1] => ? => ? = 10
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,4,3,6,5,7,2,8,10,1] => ? => ? = 10
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [2,1,9,5,4,7,6,8,10,3] => ? => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [3,2,9,5,4,7,6,8,10,1] => ? => ? = 10
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,6,4,3,5,7,2,8,10,1] => ? => ? = 10
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [2,1,9,6,5,7,4,8,10,3] => ? => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [3,2,9,6,5,7,4,8,10,1] => ? => ? = 10
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,4,1,6,5,9,8,10,7] => ? => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,2,4,6,1,9,8,10,7] => ? => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3,9,8,10,7] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1,9,8,10,7] => ? => ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [2,1,7,5,4,6,9,8,10,3] => ? => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [3,2,7,5,4,6,9,8,10,1] => ? => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,3,2,5,4,6,9,8,10,1] => ? => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,4,3,5,2,6,9,8,10,1] => ? => ? = 10
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,4,1,7,6,9,8,10,5] => ? => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,9,8,10,1] => ? => ? = 10
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7,10,9,12,11] => ? => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7,10,9,12,11] => ? => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7,10,9,12,11] => ? => ? = 2

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: 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: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 91%

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
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => [8,2,1,5,4,6,3,7] => ?
 => ? = 8 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7] => [6,3,2,4,1,5,8,7] => ?
 => ? = 6 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => [8,3,2,4,1,6,5,7] => ?
 => ? = 8 + 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => [8,3,2,5,4,6,1,7] => ?
 => ? = 8 + 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [8,3,7,6,5,4,2,1] => [8,7,2,6,5,4,3,1] => ?
 => ? = 8 + 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [8,4,3,7,6,5,2,1] => [8,7,3,2,6,5,4,1] => ?
 => ? = 8 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [8,7,4,6,5,3,2,1] => [8,7,6,3,5,4,2,1] => ?
 => ? = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7,10,9] => [4,2,1,3,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,5,4,6,1,8,7,10,9] => [6,2,1,4,3,5,8,7,10,9] => ?
 => ? = 6 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,5,4,7,6,8,1,10,9] => [8,2,1,4,3,6,5,7,10,9] => ?
 => ? = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,6,5,9,8,10,7] => [2,1,4,3,6,5,10,8,7,9] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,9,8,10,5] => [2,1,4,3,10,6,5,8,7,9] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,9,8,10,3] => [2,1,10,4,3,6,5,8,7,9] => ?
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7,10,9] => ? => ?
 => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7,10,9] => [2,1,6,4,3,5,8,7,10,9] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,1,7,5,4,6,8,3,10,9] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [3,2,7,5,4,6,8,1,10,9] => ? => ?
 => ? = 8 + 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,3,2,5,4,6,8,1,10,9] => ? => ?
 => ? = 8 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,4,3,5,2,6,8,1,10,9] => ? => ?
 => ? = 8 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,4,3,7,6,8,5,10,9] => [2,1,4,3,8,6,5,7,10,9] => ?
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,4,1,7,6,8,5,10,9] => [4,2,1,3,8,6,5,7,10,9] => ?
 => ? = 4 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,7,6,8,1,10,9] => ? => ?
 => ? = 8 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,5,4,7,6,8,3,10,9] => [2,1,8,4,3,6,5,7,10,9] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,4,3,9,7,6,8,10,5] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,4,1,9,7,6,8,10,5] => ? => ?
 => ? = 4 + 1
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,2,4,9,7,6,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,5,4,9,7,6,8,10,3] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,5,4,9,7,6,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,3,2,5,4,7,6,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [9,4,3,5,2,7,6,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,3,2,6,5,7,4,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,4,3,6,5,7,2,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [2,1,9,5,4,7,6,8,10,3] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [3,2,9,5,4,7,6,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [9,6,4,3,5,7,2,8,10,1] => [10,7,4,3,5,2,6,8,1,9] => ?
 => ? = 10 + 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [2,1,9,6,5,7,4,8,10,3] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [3,2,9,6,5,7,4,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,4,1,6,5,9,8,10,7] => [4,2,1,3,6,5,10,8,7,9] => ?
 => ? = 4 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,2,4,6,1,9,8,10,7] => ? => ?
 => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3,9,8,10,7] => [2,1,6,4,3,5,10,8,7,9] => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1,9,8,10,7] => [6,2,1,4,3,5,10,8,7,9] => ?
 => ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [2,1,7,5,4,6,9,8,10,3] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [3,2,7,5,4,6,9,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,3,2,5,4,6,9,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,4,3,5,2,6,9,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,4,1,7,6,9,8,10,5] => [4,2,1,3,10,6,5,8,7,9] => ?
 => ? = 4 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,9,8,10,1] => ? => ?
 => ? = 10 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7,10,9,12,11] => [4,2,1,3,6,5,8,7,10,9,12,11] => ?
 => ? = 4 + 1

Description

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

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: 73% ●values known / values provided: 78%●distinct values known / distinct values provided: 73%

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
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [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
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,9] => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ?
 => ? = 10
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [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
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [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
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [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
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [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
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [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
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9] => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 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] => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2,2,2] => [1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2,2] => [1,2,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2] => [1,2,2,1,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,2,2,4] => [1,1,1,2,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,2,6] => [1,1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,8] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2] => [1,2,2,1,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,2,2] => [1,2,2,1,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [2,6,2] => [1,2,1,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,2,4,2] => [1,2,1,1,2,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2] => [1,2,1,1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,2] => [1,2,1,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,2,6] => [1,1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,6] => [1,1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,8] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [2,8] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [2,8] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,4] => [1,1,1,2,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4] => [1,1,1,2,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,4] => [1,1,1,2,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2

Description

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

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: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 91%

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
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2] => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [6,2] => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => [6,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [6,2] => ? = 2
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => [8] => ? = 8
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [8] => ? = 8
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [8] => ? = 8
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,4] => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,6] => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [8] => ? = 8
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [8] => ? = 8
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => [8] => ? = 8
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [8] => ? = 8
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6] => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,6] => ? = 6
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8] => ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [8] => ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2,2,2] => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,2,4] => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,6] => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [2,8] => ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [10] => ? = 10
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2] => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [8,2] => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,6] => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,2,4,2] => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => [2,6,2] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [2,8] => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [2,8] => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [2,8] => ? = 8
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [2,4,2,2] => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [2,4,4] => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [2,8] => ? = 8
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [2,6,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [6,2,2] => ? = 2
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
 => [6,4] => ? = 4
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
 => [10] => ? = 10
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
 => [8,2] => ? = 2

Description

The last part of an integer composition.

Matching statistic: St000026
(load all 39 compositions to match this statistic)

Mapping

St000026: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 73%●distinct values known / distinct values provided: 64%

Values

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

Description

The position of the first return of a Dyck path.

Matching statistic: St000025
(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
St000025: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 73%●distinct values known / distinct values provided: 64%

Values

[1,0]
 => [1] => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 3
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 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]
 => 2
[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]
 => 2
[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]
 => 2
[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]
 => 2
[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]
 => 2
[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]
 => 3
[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]
 => 3
[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]
 => 4
[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]
 => 5
[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]
 => 5
[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]
 => 4
[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]
 => 5
[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]
 => 5
[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]
 => 5
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,8,7,6,5,4,3] => [1,2,8,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,3,8,7,6,5,4,2] => [1,8,2,7,6,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,8,3,7,6,5,4,2] => [1,8,3,7,6,5,4,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,8,4,7,6,5,3,2] => [1,8,7,3,6,5,4,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,8,7,4,5,6,3,2] => [1,8,7,4,5,6,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,8,7,4,6,5,3,2] => [1,8,7,4,6,5,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,7,6,5,4,3,2,8] => [1,7,6,5,4,3,2,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,8,6,5,4,7,3,2] => [1,8,7,5,4,3,6,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,8,7,5,4,6,3,2] => [1,8,7,5,4,6,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,8,7,5,6,4,3,2] => [1,8,7,6,4,5,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,8,5] => [2,1,4,3,8,6,5,7] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7] => [2,1,6,4,3,5,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,8,3] => [2,1,8,4,3,6,5,7] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => [2,1,8,5,4,6,3,7] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,7,6,5,4,1] => [8,1,2,7,6,5,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [2,8,3,7,6,5,4,1] => [8,1,3,7,6,5,4,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,8,7,6,5,4,3,1] => [8,1,7,6,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7] => [4,2,1,3,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,4,1,7,6,8,5] => [4,2,1,3,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,5,4,6,1,8,7] => [6,2,1,4,3,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,7,6,8,1] => [8,2,1,4,3,6,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => [8,2,1,5,4,6,3,7] => ?
 => ? = 8
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,2,8,7,6,5,4,1] => [8,2,1,7,6,5,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [8,2,7,6,5,4,3,1] => [8,2,7,6,5,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7] => [6,3,2,4,1,5,8,7] => ?
 => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => [8,3,2,4,1,6,5,7] => ?
 => ? = 8
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => [8,3,2,5,4,6,1,7] => ?
 => ? = 8
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => [8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [7,3,6,5,4,2,1,8] => [7,6,2,5,4,3,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [8,3,7,6,5,4,2,1] => [8,7,2,6,5,4,3,1] => ?
 => ? = 8
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [7,4,3,5,2,6,8,1] => [8,5,3,2,4,6,1,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [8,4,3,7,6,5,2,1] => [8,7,3,2,6,5,4,1] => ?
 => ? = 8
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [7,6,3,4,5,2,1,8] => [7,6,3,4,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [7,6,3,5,4,2,1,8] => [7,6,3,5,4,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [8,7,3,6,5,4,2,1] => [8,7,3,6,5,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [7,5,4,3,6,2,1,8] => [7,6,4,3,2,5,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [7,6,4,3,5,2,1,8] => [7,6,4,3,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,6,4,5,3,2,1,8] => [7,6,5,3,4,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [8,7,4,6,5,3,2,1] => [8,7,6,3,5,4,2,1] => ?
 => ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [8,7,5,4,6,3,2,1] => [8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8,7,6,4,5,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 8

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

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

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. St000054The first entry of the permutation. St001725The harmonious chromatic number of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. 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.