- FindStat

Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000386
(load all 34 compositions to match this statistic)

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of factors DDU in a Dyck path.

Matching statistic: St000291

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 84%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of descents of a binary word.

Matching statistic: St001712

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.

Matching statistic: St001037
(load all 6 compositions to match this statistic)

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000071

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of maximal chains in a poset.

Matching statistic: St000068

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of minimal elements in a poset.

Matching statistic: St000201
(load all 7 compositions to match this statistic)

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [.,.]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[.,[[[[[[.,.],.],.],.],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [[[[[[.,[[.,.],.]],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[.,[[[[[.,[.,.]],.],.],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [[[[[.,[[[.,.],.],.]],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [[[[[.,[[.,.],[.,.]]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [.,[[[[.,[[.,.],[.,.]]],.],.],.]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],[[.,.],.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],[[.,.],.]],.],.],.]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [.,[[[[[.,.],[.,[.,.]]],.],.],.]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [.,[[[[[.,[.,.]],[.,.]],.],.],.]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [[[[[[[.,.],.],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [[[[.,[[[[.,.],.],.],.]],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[.,[[[[.,.],.],.],.]],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [.,[.,[[.,[[[[.,.],.],.],.]],.]]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [.,[[[.,[[.,[[.,.],.]],.]],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [.,[[[.,[[[.,.],[.,.]],.]],.],.]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [.,[[[.,[.,[[[.,.],.],.]]],.],.]]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [[[[.,[.,[[.,[.,.]],.]]],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [.,[[[.,[.,[[.,[.,.]],.]]],.],.]]
 => ? = 0 + 1

Description

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].

Matching statistic: St000257
(load all 2 compositions to match this statistic)

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

Matching statistic: St000527

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 80%

Values

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

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St000568

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 80%

Values

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

Description

The hook number of a binary tree. A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.

The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St001840The number of descents of a set partition. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St000647The number of big descents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000779The tier of a permutation. St000251The number of nonsingleton blocks of a set partition. St000023The number of inner peaks of a permutation. St000360The number of occurrences of the pattern 32-1. St001728The number of invisible descents of a permutation. St000092The number of outer peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000646The number of big ascents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000252The number of nodes of degree 3 of a binary tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000455The second largest eigenvalue of a graph if it is integral. St001487The number of inner corners of a skew partition.