Your data matches 54 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001176
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,1]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [5,1]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [3,1,1]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [5,1]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [6,1]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,3,4,2,1,6] => [4,1,1]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,6,4,5,3,2] => [4,1,1]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [6,1]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => [7,1]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,5,3,4,2,1,7] => [5,1,1]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,3,2,4,1,6] => [4,1,1]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,2,4,3,1,6] => [4,1,1]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,6,4,3,5,2] => [4,1,1]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,6,3,5,4,2] => [4,1,1]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,7,6,4,5,3,2] => [5,1,1]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => [7,1]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,7,6,5,4,3,2,1,9] => [8,1]
 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,6,4,5,3,2,1,8] => [6,1,1]
 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,4,3,5,2,1,7] => [5,1,1]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,3,5,4,2,1,7] => [5,1,1]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => [4,1,1]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,2,3,4,1,6] => [3,1,1,1]
 => 3
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000394
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,3,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[4,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[4,3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 3
[4,3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4
[4,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4
[4,3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[7,7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,5,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[7,3,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[6,6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[6,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[4,3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[4,3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[4,3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[3,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[9,7]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => ?
 => ? = 2
[7,7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[4,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[4,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[9,8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[9,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[8,8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,4,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
[5,4,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
[5,4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[2,2,2,2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[5,4,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[6,4,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[5,4,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 5
[5,5,5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[5,4,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[7,6,3,2,1,1]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[5,4,4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[7,6,5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[6,6,5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[7,6,4,2,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[6,6,4,2,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[7,5,4,2,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[7,6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 5
[2,2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[4,3,3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 2
[8,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000024
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St001189
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001007
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001088
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 1
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St000272
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => ([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? = 3
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [8,5,2,1,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,3,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? = 3
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [7,2,3,4,5,6,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? = 3
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
Description
The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000362
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => ([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? = 3
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [8,5,2,1,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,3,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? = 3
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [7,2,3,4,5,6,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? = 3
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
Description
The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000536
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => ([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? = 3
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [8,5,2,1,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,3,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? = 3
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [7,2,3,4,5,6,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? = 3
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
Description
The pathwidth of a graph.
Matching statistic: St001580
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001580: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 3 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 3 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => ([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? = 3 + 1
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3 + 1
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [8,5,2,1,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,3,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? = 3 + 1
[7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 + 1
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [7,2,3,4,5,6,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? = 3 + 1
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,2,7,8,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3 + 1
Description
The acyclic chromatic number of a graph. This is the smallest size of a vertex partition $\{V_1,\dots,V_k\}$ such that each $V_i$ is an independent set and for all $i,j$ the subgraph inducted by $V_i\cup V_j$ does not contain a cycle.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St001331The size of the minimal feedback vertex set. St001963The tree-depth of a graph. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000010The length of the partition. St001484The number of singletons of an integer partition. St000527The width of the poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000159The number of distinct parts of the integer partition. St000481The number of upper covers of a partition in dominance order. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000862The number of parts of the shifted shape of a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000336The leg major index of a standard tableau. St001480The number of simple summands of the module J^2/J^3. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000758The length of the longest staircase fitting into an integer composition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001029The size of the core of a graph. St000535The rank-width of a graph. St001668The number of points of the poset minus the width of the poset. St000144The pyramid weight of the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St000632The jump number of the poset. St001734The lettericity of a graph. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix.