- FindStat

Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000883
(load all 18 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of longest increasing subsequences of a permutation.

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

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leading ones in a binary word.

Matching statistic: St000326

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] =>  => ? = 1
[1,0,1,0]
 => [1,2] => [1,2] => 0 => 2
[1,1,0,0]
 => [2,1] => [2,1] => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 00 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 10 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 01 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 11 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 000 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,4,1,2] => 010 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => 110 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,1,3] => 010 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => 010 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => 001 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => 101 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 110 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 101 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => 011 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 111 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => 1000 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,5,1,2,3] => 0100 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => 1100 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,5,1,2,4] => 0100 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,4,1,2,5] => 0100 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,4,5,1,2] => 0010 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,4,1,2] => 1010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => 1100 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,5,1,2] => 1010 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,3,1,2] => 0110 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,5,1,3,4] => 0100 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,4,1,3,5] => 0100 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,5,1,3] => 0010 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,2,4,1,3] => 1010 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => 0100 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,5,1,4] => 0010 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => 0010 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => 1001 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => 1010 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => 1001 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,5,2,3,1] => 0101 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => 1101 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => 1100 => 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,5,6,4,3,2,8,7] => [5,6,4,3,8,1,2,7] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,5,6,4,7,8,1] => [5,6,2,3,4,7,8,1] => ? => ? = 2
[1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [2,4,5,3,7,8,6,1] => [4,5,7,8,2,3,6,1] => ? => ? = 4
[]
 => [] => [] => ? => ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,4,6,5,7,8,9,1] => [6,2,3,4,5,7,8,9,1] => ? => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,4,6,7,5,8,9,1] => [6,7,2,3,4,5,8,9,1] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,9,8,1] => [3,5,7,9,2,4,6,8,1] => ? => ? = 4
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,9,8,7,6,1] => [5,9,4,8,3,7,2,6,1] => ? => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,4,7,6,9,8,3,2,1] => [5,7,9,4,6,8,3,2,1] => ? => ? = 3
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [6,5,4,9,8,7,3,2,1] => [6,9,5,8,4,7,3,2,1] => ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? => ? => ? => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? => ? => ? => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,4,5,7,6,8,9,10,1] => [7,2,3,4,5,6,8,9,10,1] => ? => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => [8,7,10,9,6,5,4,3,2,1] => [8,10,7,9,6,5,4,3,2,1] => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,10,9,8,4,3,2,1] => [7,10,6,9,5,8,4,3,2,1] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,10,9,8,7,2,1] => [6,10,5,9,4,8,3,7,2,1] => ? => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
 => [8,7,6,5,4,3,2,10,9,1] => [8,7,6,5,4,3,10,2,9,1] => ? => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [4,3,6,5,8,7,10,9,2,1] => [4,6,8,10,3,5,7,9,2,1] => ? => ? = 4
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [4,3,2,7,6,5,10,9,8,1] => [4,7,10,3,6,9,2,5,8,1] => ? => ? = 3

Description

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

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

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

Matching statistic: St000382

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

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

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000617
(load all 11 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 76%●distinct values known / distinct values provided: 60%

Values

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

Description

The number of global maxima of a Dyck path.

Matching statistic: St000678

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%

Values

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

Description

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

Matching statistic: St000069

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of maximal elements of a poset.

Matching statistic: St001622

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .

Matching statistic: St000054
(load all 2 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%

Values

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

Description

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

$\pi$ of

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

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

$1$ to

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

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

$\pi$ is obtained by replacing each number

$i$ by

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

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

$\pi$ where the arc

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

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

$\pi$ , then the number of deficiencies of

$\pi$ equals

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

The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000909The number of maximal chains of maximal size in a poset. St000026The position of the first return of a Dyck path. St000363The number of minimal vertex covers of a graph. St000911The number of maximal antichains of maximal size in a poset. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001498The normalised height of a Nakayama algebra with magnitude 1. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000738The first entry in the last row of a standard tableau. St000025The number of initial rises of a Dyck path. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. 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. St000061The number of nodes on the left branch of a binary tree. St001651The Frankl number of a lattice. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001875The number of simple modules with projective dimension at most 1. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001267The length of the Lyndon factorization of the binary word. St001904The length of the initial strictly increasing segment of a parking function.