- FindStat

Your data matches 61 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000288

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 91%

Values

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

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000394
(load all 3 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St000476
(load all 6 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 73%

Values

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

Description

The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley

$v$ in a Dyck path

$D$ there is a corresponding tunnel, which is the factor

$T_v = s_i\dots s_j$ of

$D$ where

$s_i$ is the step after the first intersection of

$D$ with the line

$y = ht(v)$ to the left of

$s_j$ . This statistic is

$\sum_v (j_v-i_v)/2.$

Matching statistic: St000147
(load all 6 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 73%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000093

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%

Values

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

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000786

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 64%

Values

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

Description

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

Matching statistic: St000024

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000987
(load all 15 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.

Matching statistic: St001189

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000019The cardinality of the support of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000740The last entry of a permutation. St000653The last descent of a permutation. St000957The number of Bruhat lower covers of a permutation. St000809The reduced reflection length of the permutation. St001497The position of the largest weak excedence of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000067The inversion number of the alternating sign matrix. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000242The number of indices that are not cyclical small weak excedances. St000358The number of occurrences of the pattern 31-2. St000673The number of non-fixed points of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000327The number of cover relations in a poset. St001861The number of Bruhat lower covers of a permutation. St001645The pebbling number of a connected graph. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000133The "bounce" of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001960The number of descents of a permutation minus one if its first entry is not one. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .