Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001483
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 5
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 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]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 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]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => 5
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 4
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 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]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => 4
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => 5
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 4
[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]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000731
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2 = 3 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0 = 1 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 3 = 4 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0 = 1 - 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]
 => [1,2,3,5,4] => 0 = 1 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => 4 = 5 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2 = 3 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1 = 2 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0 = 1 - 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]
 => [1,2,3,4,6,5] => 0 = 1 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => 5 = 6 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => 3 = 4 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1 = 2 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 2 = 3 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 3 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1 = 2 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1 = 2 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 0 = 1 - 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]
 => [1,2,3,4,5,7,6] => 0 = 1 - 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,1] => 4 = 5 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => 2 = 3 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 3 = 4 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => 0 = 1 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 1 = 2 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 2 = 3 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 0 = 1 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,3,1,5,6] => 1 = 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]
 => [1,2,5,4,3] => 0 = 1 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 0 = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7] => 0 = 1 - 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [4,2,3,5,6,7,1] => 3 = 4 - 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,3,5,6,7,1] => ? = 5 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,2,3,4,6,1] => 1 = 2 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => 2 = 3 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 3 = 4 - 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]
 => [1,3,4,5,6,2] => 3 = 4 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1 = 2 - 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [2,4,3,1,5,6,7] => ? = 2 - 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [3,2,5,4,6,7,1] => ? = 4 - 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,5,4,6,7,1] => ? = 5 - 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,3,5,4,1,6,7] => ? = 3 - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => ? = 1 - 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [4,2,3,6,5,7,1] => ? = 3 - 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,3,4,6,7,1] => ? = 4 - 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [3,2,4,6,5,7,1] => ? = 4 - 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,5,7,1] => ? = 5 - 1
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,4,6,5,1,7] => ? = 4 - 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1,6,5,4,7] => ? = 2 - 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => ? = 1 - 1
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [7,2,3,4,5,6,1] => ? = 1 - 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [5,2,3,4,7,6,1] => ? = 2 - 1
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,2,6,4,5,7,1] => ? = 3 - 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [4,2,3,5,7,6,1] => ? = 3 - 1
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,3,6,5,7,1] => ? = 4 - 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [3,2,4,5,7,6,1] => ? = 4 - 1
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => ? = 5 - 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [2,3,4,1,7,6,5] => ? = 3 - 1
[4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,5,3,7] => ? = 2 - 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,7,6,5] => ? = 2 - 1
[3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,2,3,1,5,6,7] => ? = 1 - 1
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? = 1 - 1
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [7,3,2,4,5,6,1] => ? = 1 - 1
[6,4,2]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,2,3,7,5,6,1] => ? = 2 - 1
[6,4,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 2 - 1
[6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [2,6,3,4,5,7,1] => ? = 3 - 1
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,1] => ? = 3 - 1
[6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,6,4,5,7,1] => ? = 4 - 1
[6,2,2,1,1]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [2,4,3,5,7,6,1] => ? = 4 - 1
[5,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,5,3,4,6,7,2] => ? = 3 - 1
[5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,1,5,7,6,4] => ? = 3 - 1
[5,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [4,3,2,5,6,1,7] => ? = 3 - 1
[4,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,6,5] => ? = 1 - 1
[4,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [2,5,3,4,1,6,7] => ? = 2 - 1
[4,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [4,3,2,5,1,6,7] => ? = 2 - 1
[3,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,5,3,4,2,6,7] => ? = 1 - 1
[6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [7,3,4,2,5,6,1] => ? = 2 - 1
[6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [7,2,4,3,5,6,1] => ? = 1 - 1
[6,4,3]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [3,2,7,4,5,6,1] => ? = 2 - 1
[6,4,2,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [5,3,2,4,7,6,1] => ? = 2 - 1
[6,4,1,1,1]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [6,2,4,3,5,7,1] => ? = 2 - 1
[6,3,3,1]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [2,5,3,4,7,6,1] => ? = 3 - 1
[6,3,2,2]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [3,2,4,7,5,6,1] => ? = 3 - 1
[6,3,2,1,1]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [4,3,2,6,5,7,1] => ? = 3 - 1
[6,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [5,2,4,3,6,7,1] => ? = 3 - 1
[6,2,2,2,1]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,5,4,7,6,1] => ? = 4 - 1
[6,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,5,4,3,6,7,1] => ? = 4 - 1
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000039
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2 = 3 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0 = 1 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 3 = 4 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0 = 1 - 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]
 => [1,2,3,5,4] => 0 = 1 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => 4 = 5 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2 = 3 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1 = 2 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0 = 1 - 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]
 => [1,2,3,4,6,5] => 0 = 1 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => ? = 6 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => 3 = 4 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1 = 2 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 2 = 3 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 3 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1 = 2 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1 = 2 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 0 = 1 - 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]
 => [1,2,3,4,5,7,6] => ? = 1 - 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,1] => ? = 5 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => 2 = 3 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 3 = 4 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => 0 = 1 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 1 = 2 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 2 = 3 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 0 = 1 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,3,1,5,6] => 1 = 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]
 => [1,2,5,4,3] => 0 = 1 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 0 = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7] => ? = 1 - 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [4,2,3,5,6,7,1] => ? = 4 - 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,3,5,6,7,1] => ? = 5 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,2,3,4,6,1] => 1 = 2 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => 2 = 3 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 3 = 4 - 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]
 => [1,3,4,5,6,2] => 3 = 4 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1 = 2 - 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,4,1,6] => 2 = 3 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 0 = 1 - 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0 = 1 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 0 = 1 - 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [2,4,3,1,5,6,7] => ? = 2 - 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,4,3,2,5,6,7] => ? = 1 - 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [5,2,3,4,6,7,1] => ? = 3 - 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [3,2,5,4,6,7,1] => ? = 4 - 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,5,4,6,7,1] => ? = 5 - 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,3,5,4,1,6,7] => ? = 3 - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => ? = 1 - 1
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => ? = 1 - 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [6,2,3,4,5,7,1] => ? = 2 - 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [4,2,3,6,5,7,1] => ? = 3 - 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,3,4,6,7,1] => ? = 4 - 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [3,2,4,6,5,7,1] => ? = 4 - 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,5,7,1] => ? = 5 - 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => ? = 5 - 1
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,4,6,5,1,7] => ? = 4 - 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1,6,5,4,7] => ? = 2 - 1
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,3,5,4,2,6,7] => ? = 2 - 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => ? = 1 - 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => ? = 2 - 1
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => ? = 1 - 1
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [7,2,3,4,5,6,1] => ? = 1 - 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [5,2,3,4,7,6,1] => ? = 2 - 1
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,2,6,4,5,7,1] => ? = 3 - 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [4,2,3,5,7,6,1] => ? = 3 - 1
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,3,6,5,7,1] => ? = 4 - 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [3,2,4,5,7,6,1] => ? = 4 - 1
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => ? = 5 - 1
[5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,3,5,6,7,2] => ? = 4 - 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [2,3,4,1,7,6,5] => ? = 3 - 1
[4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,5,3,7] => ? = 2 - 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,7,6,5] => ? = 2 - 1
[3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,5,4,7] => ? = 1 - 1
[3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,2,3,1,5,6,7] => ? = 1 - 1
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? = 1 - 1
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 1 - 1
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [7,3,2,4,5,6,1] => ? = 1 - 1
[6,4,2]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,2,3,7,5,6,1] => ? = 2 - 1
[6,4,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 2 - 1
[6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [2,6,3,4,5,7,1] => ? = 3 - 1
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,1] => ? = 3 - 1
[6,3,1,1,1]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [5,3,2,4,6,7,1] => ? = 3 - 1
[6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,6,4,5,7,1] => ? = 4 - 1
[6,2,2,1,1]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [2,4,3,5,7,6,1] => ? = 4 - 1
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [4,3,2,5,6,7,1] => ? = 4 - 1
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St001232
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 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]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? = 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 5
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 4
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 2
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 5
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4
[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]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 3
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 3
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 4
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 5
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 2
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 3
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.