Processing math: 85%

Your data matches 48 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000010
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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] => [1]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 3
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 4
Description
The length of the partition.
Matching statistic: St001250
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1,0]
 => [1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 3
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 4
[]
 => []
 => [] => []
 => ? = 0
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St000507
(load all 5 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [[1]]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [[1],[2]]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [[1,2]]
 => 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [[1,2],[3]]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [[1,2],[3]]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [[1,3],[2]]
 => 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [[1,3],[2]]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [[1,2,4],[3]]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [[1,2,3,4],[5]]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [[1,3,4],[2,5]]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[1,3,5],[2,4]]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [[1,2,3],[4,5]]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [[1,2,3,5],[4]]
 => 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,5,2,6,7,8,4] => ?
 => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,4,2,5,3,7,8,6] => ?
 => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,6,3,7,8,5] => ?
 => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,3,2,6,4,7,8,5] => ?
 => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,3,5,2,4,7,8,6] => ?
 => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [3,1,5,2,4,7,8,6] => ?
 => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [2,4,6,1,3,7,8,5] => ?
 => ? = 6
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,5,7,2,8,6] => ?
 => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,7,4,8,6] => ?
 => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,6,7,3,8,5] => ?
 => ? = 6
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,3,4,5,2,6,8,7] => ?
 => ? = 6
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,4,5,6,2,3,8,7] => ?
 => ? = 6
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [2,4,5,7,1,3,8,6] => ?
 => ? = 6
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [2,3,6,7,1,4,8,5] => ?
 => ? = 6
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [3,4,1,7,2,5,8,6] => ?
 => ? = 5
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
 => [4,6,1,7,2,3,8,5] => ?
 => ? = 5
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [3,1,5,2,4,6,8,7] => ?
 => ? = 5
[1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,5,7,1,2,3,8,6] => ?
 => ? = 6
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [2,4,5,1,6,7,3,8] => ?
 => ? = 6
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [2,1,3,4,6,7,5,8] => ?
 => ? = 6
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [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
[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [3,1,2,5,7,8,4,6] => ?
 => ? = 6
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,6,1,7,8,3,5] => ?
 => ? = 6
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
 => [5,1,6,2,7,8,3,4] => ?
 => ? = 5
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,4,5,6,2,8,3,7] => ?
 => ? = 6
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [2,5,1,6,3,8,4,7] => ?
 => ? = 5
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [3,1,4,7,2,8,5,6] => ?
 => ? = 5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [2,4,6,7,1,8,3,5] => ?
 => ? = 6
[1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [2,1,6,7,3,8,4,5] => ?
 => ? = 5
[1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [2,5,1,7,3,8,4,6] => ?
 => ? = 5
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,4] => ?
 => ? = 5
[1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [2,4,7,1,3,8,5,6] => ?
 => ? = 6
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [2,5,7,1,3,8,4,6] => ?
 => ? = 6
[1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [3,5,7,1,2,8,4,6] => ?
 => ? = 6
[1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,5,6,4,7,8] => ?
 => ? = 6
[1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,3,5,2,6,4,7,8] => ?
 => ? = 6
[1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,4,6,7,3,5,8] => ?
 => ? = 6
[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,6,7,3,5,8] => ?
 => ? = 7
[1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [2,4,1,6,7,3,5,8] => ?
 => ? = 6
[1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [3,4,5,1,7,2,6,8] => ?
 => ? = 6
[1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
 => [1,5,6,2,7,3,4,8] => ?
 => ? = 6
[1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [5,6,1,2,7,3,4,8] => ?
 => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
 => [4,1,5,7,2,3,6,8] => ?
 => ? = 6
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,6,7,1,4,5,8] => ?
 => ? = 7
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => [4,5,1,7,2,3,6,8] => ?
 => ? = 6
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => [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
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,1,5,2,4,6,7,8] => ?
 => ? = 6
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
 => [1,5,7,2,3,4,6,8] => ?
 => ? = 7
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => [[1,3,4,5,6,7,8],[2,9]]
 => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 10
Description
The number of ascents of a standard tableau. Entry i of a standard Young tableau is an '''ascent''' if i+1 appears to the right or above i in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000393
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1,1,0,0]
 => [2] => 10 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => 101 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => 100 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 1011 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 1010 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 1001 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 1001 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1000 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 10111 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 10110 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 10100 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => 10011 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => 10010 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => 10011 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => 10011 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => 10010 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => 10001 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => 10001 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => 10001 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => 10000 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => 101111 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => 101110 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => 101101 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => 101101 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => 101100 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => 101011 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => 101010 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => 101011 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => 101011 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => 101010 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => 101001 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => 101001 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => 101001 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => 101000 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => 100111 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => 100110 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => 100101 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => 100101 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => 100100 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => 100111 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => 100110 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => 100111 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => 100111 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => 100110 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => 100101 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => 100101 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => 100101 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => 100100 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? => ? => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => ? => ? => ? = 5 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? => ? => ? = 5 + 1
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000676
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [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] => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,1,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,6,1,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,5,1,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,4,5,7,1,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,1,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,4,5,7,1,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,4,5,1,8,6,7] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,5,8,1,6,7] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,4,1,6,7,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5,8,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,3,4,1,6,8,5,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,4,1,6,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,3,4,6,1,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 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] => ?
 => ?
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,6,7,8,1,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,7,1,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 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] => ?
 => ?
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,1,8,5,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,4,6,8,1,5,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,6,1,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,3,4,1,5,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,4,1,5,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,4,1,7,5,8,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,4,1,7,8,5,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,4,1,7,5,6,8] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,4,7,1,5,8,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,4,7,1,8,5,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,4,7,8,1,5,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6,8] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,4,1,5,8,6,7] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,4,1,8,5,6,7] => [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,8,4] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,3,1,5,6,7,4,8] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 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] => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,1,5,6,4,7,8] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,7,6,8] => [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1,5,7,4,8,6] => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,1,5,7,4,6,8] => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1,5,4,6,8,7] => ?
 => ?
 => ? = 5
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength n with k up steps in odd positions and k returns to the main diagonal are counted by the binomial coefficient \binom{n-1}{k-1} [3,4].
Matching statistic: St000245
(load all 9 compositions to match this statistic)
Mapping
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [1,0]
 => [[1]]
 => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [5,1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [4,1,2,3,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [4,1,2,3,7,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [4,1,2,3,6,5,7] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [4,1,2,3,7,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [4,1,2,3,7,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [4,1,2,3,6,5,7] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,7,4,5,6] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,6,4,5,7] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [3,1,2,5,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [3,1,2,5,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [3,1,2,5,4,6,7] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,7,4,5,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,6,4,5,7] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,7,4,5,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,7,4,5,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,6,4,5,7] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [3,1,2,5,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [3,1,2,5,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [3,1,2,5,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [3,1,2,5,4,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,4,6,5,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,4,6,5,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,4,6,5,7] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,5,3,4,6,7] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,5,6,7] => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,5,3,4,6,7] => ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,5,3,4,6,7] => ? = 5 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 4 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000250
(load all 2 compositions to match this statistic)
Mapping
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => [1,0]
 => {{1}}
 => ? = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 2 = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 4 = 3 + 1
[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},{2},{3},{4}}
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 4 = 3 + 1
[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},{2,3,4}}
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 4 = 3 + 1
[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]
 => {{1,3,5},{2},{4}}
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 4 = 3 + 1
[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]
 => {{1},{2,3,5},{4}}
 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 5 = 4 + 1
[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},{2,4,5},{3}}
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 5 = 4 + 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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [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]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 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]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1},{2,8},{3},{4},{5},{6},{7}}
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => {{1,7},{2},{3},{4},{5},{6},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [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]
 => {{1,2,8},{3},{4},{5},{6},{7}}
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1},{2},{3,8},{4},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => {{1,3,8},{2},{4},{5},{6},{7}}
 => ? = 6 + 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]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => {{1},{2,7},{3},{4},{5},{6},{8}}
 => ? = 6 + 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]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => {{1,6},{2},{3},{4},{5},{7},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 6 + 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]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => {{1},{2,3,8},{4},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => {{1,2,7},{3},{4},{5},{6},{8}}
 => ? = 6 + 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]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,2},{3,8},{4},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5},{6},{7}}
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2},{3},{4,8},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [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]
 => {{1,4,8},{2},{3},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => {{1},{2,4,8},{3},{5},{6},{7}}
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => {{1,3,7},{2},{4},{5},{6},{8}}
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,4,8},{3},{5},{6},{7}}
 => ? = 6 + 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]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => {{1},{2},{3,7},{4},{5},{6},{8}}
 => ? = 6 + 1
[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]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => {{1,8},{2},{3,7},{4},{5},{6}}
 => ? = 6 + 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]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => {{1},{2,6},{3},{4},{5},{7},{8}}
 => ? = 6 + 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]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => {{1,5},{2},{3},{4},{6},{7},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => {{1,8},{2,6},{3},{4},{5},{7}}
 => ? = 6 + 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]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => {{1},{2,8},{3,7},{4},{5},{6}}
 => ? = 5 + 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]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => {{1,7},{2,6},{3},{4},{5},{8}}
 => ? = 5 + 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]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => {{1,3},{2},{4,8},{5},{6},{7}}
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => {{1,2,8},{3,7},{4},{5},{6}}
 => ? = 6 + 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]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => {{1},{2},{3,4,8},{5},{6},{7}}
 => ? = 6 + 1
[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]
 => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,3,4,8},{2},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => {{1},{2,3,7},{4},{5},{6},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => {{1,2,6},{3},{4},{5},{7},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => {{1,8},{2,3,7},{4},{5},{6}}
 => ? = 6 + 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]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,3},{4,8},{5},{6},{7}}
 => ? = 6 + 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]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1,2},{3},{4,8},{5},{6},{7}}
 => ? = 6 + 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]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => {{1,2},{3,7},{4},{5},{6},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => {{1,4,8},{2,3},{5},{6},{7}}
 => ? = 6 + 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]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => {{1},{2,3,4,8},{5},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => {{1,2,3,7},{4},{5},{6},{8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => {{1,2},{3,4,8},{5},{6},{7}}
 => ? = 6 + 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]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
 => {{1,2,6},{3},{4},{5},{7,8}}
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5},{6},{7}}
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [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]
 => {{1,5,8},{2},{3},{4},{6},{7}}
 => ? = 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]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => {{1,4,7},{2},{3},{5},{6},{8}}
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [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]
 => {{1,2,5,8},{3},{4},{6},{7}}
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => {{1,3,5,8},{2},{4},{6},{7}}
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => {{1},{2,4,7},{3},{5},{6},{8}}
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
 => {{1,8},{2,4,7},{3},{5},{6}}
 => ? = 5 + 1
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St001211
(load all 23 compositions to match this statistic)
Mapping
St001211: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 58%
Values
[1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 5 + 1
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 6 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 58%
Values
[1,0]
 => [1] => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [2] => [1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 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] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 5 + 1
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St001007
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 58%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,1,8,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,6,1,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,5,1,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,4,5,7,1,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,1,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,4,5,7,1,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,4,5,1,8,6,7] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,5,8,1,6,7] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,4,1,6,7,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5,8,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,3,4,1,6,8,5,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,4,1,6,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,3,4,6,1,7,8,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 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] => ?
 => ?
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,6,7,8,1,5] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,7,1,5,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 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] => ?
 => ?
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,1,8,5,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,4,6,8,1,5,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,6,1,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,3,4,1,5,7,8,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,4,1,5,7,6,8] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,4,1,7,5,8,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,4,1,7,8,5,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,4,1,7,5,6,8] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,4,7,1,5,8,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,4,7,1,8,5,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,4,7,8,1,5,6] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6,8] => [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 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] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,4,1,5,8,6,7] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,4,1,8,5,6,7] => [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,8,4] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,3,1,5,6,7,4,8] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 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] => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,1,5,6,4,7,8] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,7,6,8] => [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1,5,7,4,8,6] => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,1,5,7,4,6,8] => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1,5,4,6,8,7] => ?
 => ?
 => ? = 5
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000702The number of weak deficiencies of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000167The number of leaves of an ordered tree. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000308The height of the tree associated to a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001896The number of right descents of a signed permutations. St000331The number of upper interactions of a Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001626The number of maximal proper sublattices of a lattice.