Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001182
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 3
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001255
(load all 10 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 5
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 6
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000235
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [2,1] => [1,2] => 2 = 3 - 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [2,3,1] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => [3,1,2] => 3 = 4 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [3,2,1] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [2,4,3,1] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,4,2,1] => [1,4,2,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => [1,4,3,2] => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1,4,2] => [3,1,2,4] => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,1,4] => [4,2,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [2,3,4,5,1] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [2,3,5,4,1] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,4,5,3,1] => [1,2,5,3,4] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,5,4,3,1] => [1,2,5,4,3] => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [2,5,3,4,1] => [1,2,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,4,2,5,1] => [1,4,2,3,5] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,5,2,4,1] => [1,4,2,5,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,3,5,2,1] => [1,5,3,2,4] => 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,4,2,1] => [1,5,3,4,2] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,5,2,3,1] => [1,4,5,2,3] => 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,4,2,3,1] => [1,4,5,3,2] => 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => [1,4,3,5,2] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,1,4,5,2] => [3,1,2,4,5] => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,5,4,2] => [3,1,2,5,4] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,1,5,3,2] => [3,1,5,2,4] => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,1,4,3,2] => [3,1,5,4,2] => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,3,4,2] => [3,1,4,5,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,4,1,5,3] => [4,2,1,3,5] => 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [2,5,1,4,3] => [4,2,1,5,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [2,3,5,1,4] => [5,2,3,1,4] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => [5,2,3,4,1] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [2,4,3,1,5] => [5,2,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,5,2,1,4] => [5,4,2,1,3] => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,4,2,1,5] => [5,4,2,3,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => 2 = 3 - 1
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St001279
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1]
 => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [2,1] => [2]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,1,2] => [3]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [2,1]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [3]
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,1,2,3] => [4]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,1,3,2] => [3,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,3,1,2] => [4]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => [2,2]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [2,4,1,3] => [4]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,3,1] => [3,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => [4]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1,2,4] => [3,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [4]
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => [3,1]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,1,2,3,4] => [5]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,1,2,4,3] => [4,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,1,4,2,3] => [5]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,1,4,3,2] => [3,2]
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,1,3,4,2] => [3,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,3,1,2,4] => [5]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,3,1,4,2] => [4,1]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,4,2,1,3] => [5]
 => 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,4,2,3,1] => [3,2]
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,4,3,2,1] => [2,2,1]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,3,4,1,2] => [5]
 => 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,3,4,2,1] => [3,2]
 => 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => [2,2,1]
 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [2,5,1,3,4] => [5]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [2,5,1,4,3] => [4,1]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [2,5,4,1,3] => [5]
 => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [2,5,4,3,1] => [3,2]
 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,5,3,4,1] => [3,1,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,1,5,2,4] => [5]
 => 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,5,4,2] => [4,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1,2,5,3] => [5]
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,1,3,2,5] => [3,1,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,3,1,5,2] => [5]
 => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,3,1,2,5] => [4,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => [2,2,1]
 => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
 => 2 = 3 - 1
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000673
(load all 9 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ? = 1 - 1
[1,0,1,0]
 => [2,1] => [2,1] => 2 = 3 - 1
[1,1,0,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [2,3,1] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 3 = 4 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [2,4,3,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,4,2,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1,4,2] => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,1,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [2,3,4,5,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [2,3,5,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,4,5,3,1] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,5,4,3,1] => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [2,5,3,4,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,4,2,5,1] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,5,2,4,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,3,5,2,1] => 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,4,2,1] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,5,2,3,1] => 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,4,2,3,1] => 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,1,4,5,2] => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,5,4,2] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,1,5,3,2] => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,1,4,3,2] => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,3,4,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,4,1,5,3] => 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [2,5,1,4,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [2,3,5,1,4] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [2,4,3,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,5,2,1,4] => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,4,2,1,5] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,1,2,5,3] => 5 = 6 - 1
Description
The number of non-fixed points of a permutation. In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001005
(load all 3 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1] => ? = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 2 = 3 - 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3 = 4 - 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 5 = 6 - 1
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
Matching statistic: St000896
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [1] => [1] => [[1]]
 => ? = 1 - 1
[1,0,1,0]
 => [2,1] => [2,1] => [[0,1],[1,0]]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,2] => [1,2] => [[1,0],[0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,3,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
 => 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,4,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,1,5,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
 => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,1,4,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,3,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,5,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [2,4,5,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [2,4,6,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [2,5,4,6,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [2,6,4,5,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [2,6,5,4,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [2,5,6,3,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [2,6,5,3,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [2,6,4,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [2,6,3,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [3,4,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [3,4,2,6,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [3,6,2,4,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [4,3,5,2,6,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,6,3,2,4,1] => [4,3,6,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [5,3,4,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [6,3,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,6,1] => [6,3,5,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => [5,4,6,3,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => [6,4,5,3,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => [6,5,4,3,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => [6,5,3,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [4,5,2,3,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [4,6,2,3,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,5,1] => [5,4,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,6,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => [6,5,2,4,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,5,1] => [5,3,6,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [6,3,5,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [6,3,4,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [6,4,3,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [5,6,2,3,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [6,5,2,3,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [6,4,2,3,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [6,3,2,4,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 5 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [3,1,4,5,6,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [3,1,4,6,5,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [3,1,5,6,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,1,2] => [3,1,6,5,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 7 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => [3,1,6,4,5,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,1,2] => [4,1,5,3,6,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,1,2] => [4,1,6,3,5,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 6 - 1
Description
The number of zeros on the main diagonal of an alternating sign matrix.