Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001182
(load all 6 compositions to match this statistic)
Mapping
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]
 => 3
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => 3
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,1,0,1,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => 3
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000235
(load all 3 compositions to match this statistic)
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [1,2] => 2 = 3 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [3,2,1] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [2,1,3] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,4,3,2] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,2,4,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,1,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,2,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,2,5,4,3] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,2,4,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,2,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,5,3,4,2] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,4,3,5,2] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,4,3,2,5] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,5,4,2] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,3,4,2,5] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,2,3,4,1] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [4,2,3,5,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,2,5,4,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,2,4,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,2,4,5,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [2,1,3,5,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,4,5] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [4,2,3,1,5] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [3,2,1,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [2,1,4,3,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [3,2,4,1,5] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,1,4,5,3] => 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
(load all 2 compositions to match this statistic)
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [1]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2] => [2]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => [1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => [3]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => [2,1]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [2,1]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [3,2]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [4,1]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [4,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,1,1,1]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [4,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [3,2]
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => 2 = 3 - 1
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001504
(load all 2 compositions to match this statistic)
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000673
(load all 4 compositions to match this statistic)
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => ? = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 2 = 3 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[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]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 2 = 3 - 1
[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]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [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]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3 = 4 - 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: St000896
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [[1]]
 => ? = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [[0,1],[1,0]]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [[1,0],[0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [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]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [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]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [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]
 => [1,1,0,1,0,1,0,1,0,0]
 => [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]
 => [1,1,0,1,0,1,0,0,1,0]
 => [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
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [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
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [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
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [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
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[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]]
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [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
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [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
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [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
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [[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]]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [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
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [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
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [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
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [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
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [2,3,1,6,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [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
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,5,6,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [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
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [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
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 7 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,5,3,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6,3,5,4,1] => [[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],[0,1,0,0,0,0]]
 => ? = 6 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [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
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,1,6,4,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 5 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [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
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [[1,0,0,0,0,0],[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]]
 => ? = 6 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 6 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 5 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,6,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [5,3,4,2,6,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [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
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 6 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [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
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [3,2,1,5,6,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 6 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [6,3,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 7 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [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
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 5 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,6,4,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 4 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [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
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [[1,0,0,0,0,0],[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]]
 => ? = 5 - 1
Description
The number of zeros on the main diagonal of an alternating sign matrix.
Matching statistic: St000831
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [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]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,4,2,6,1] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,5,3,2,6,1] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [3,4,2,1,5,6] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,4,6,3,1] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,4,3,5,1,6] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,4,6,2,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,5,3,6,2,1] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [3,4,2,5,1,6] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,2,5,6,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,5,3,2,6] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,3,6,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,4,3,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,4,5,3,1,6] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,6,4,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,5,6,3,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,2,1,6] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,3,6,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,2,6,1] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,1,5,6] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [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]
 => [6,7,5,4,3,2,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,4,3,2,1,7] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [4,6,5,3,2,7,1] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,4,3,2,7,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [4,5,3,2,1,6,7] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [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]
 => [3,6,5,4,7,2,1] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [3,5,4,2,6,1,7] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [4,6,5,3,7,2,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,4,3,7,2,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [4,5,3,2,6,1,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [3,4,2,1,6,7,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [3,5,4,2,6,7,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [4,5,3,2,6,7,1] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [3,4,2,1,5,6,7] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [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]
 => [2,6,5,7,4,3,1] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [2,5,4,6,3,1,7] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,3,6,5,7,1] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,5,4,6,3,7,1] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,4,3,5,1,6,7] => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [3,6,5,7,4,2,1] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [3,5,4,6,2,1,7] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,6,5,7,3,2,1] => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,4,7,3,2,1] => ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [4,5,3,6,2,1,7] => ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [3,4,2,6,5,7,1] => ? = 6 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [3,5,4,6,2,7,1] => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [4,5,3,6,2,7,1] => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [3,4,2,5,1,6,7] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1,6,7,5,4] => ? = 6 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1,5,6,4,7] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,3,6,7,5,1] => ? = 6 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [2,5,4,6,7,3,1] => ? = 7 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [2,4,3,5,6,1,7] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [3,4,2,6,7,5,1] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [3,5,4,6,7,2,1] => ? = 6 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,3,6,7,2,1] => ? = 6 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [3,4,2,5,6,1,7] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [2,3,1,4,6,7,5] => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,6,7,4] => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,3,5,6,7,1] => ? = 5 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [3,4,2,5,6,7,1] => ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7] => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [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]
 => [1,6,7,5,4,3,2] => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,5,6,4,3,2,7] => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,5,6,4,3,7,2] => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,4,6,5,7,3,2] => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,5,6,4,7,3,2] => ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [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]
 => [2,6,7,5,4,3,1] => ? = 6 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [2,5,6,4,3,1,7] => ? = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,4,6,5,3,7,1] => ? = 6 - 1
Description
The number of indices that are either descents or recoils. This is, for a permutation $\pi$ of length $n$, this statistics counts the set $$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$
Matching statistic: St000141
(load all 2 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => [4,1,3,2] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,1,3,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => [5,1,4,3,2] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,4,1,2,5] => [4,1,3,2,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,2,1,4] => [5,2,3,1,4] => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,5,1,4,2] => [4,5,1,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,2,5,1,3] => [2,5,1,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,1,3,5] => [2,4,1,3,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,5,3,1,2] => [3,5,1,4,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,1,3,2] => [5,1,3,4,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [4,1,3,2,5] => [3,4,1,2,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,2,1,4,3] => [2,4,5,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,1,3,4,2] => [3,4,5,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => [6,1,5,4,3,2] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,5,1,2,6] => [5,1,4,3,2,6] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,4,6,2,1,5] => [6,2,4,3,1,5] => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,4,6,1,5,2] => [5,6,1,4,3,2] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,4,1,2,5,6] => [4,1,3,2,5,6] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,5,2,6,1,4] => [6,2,3,1,5,4] => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,5,2,1,4,6] => [5,2,3,1,4,6] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [3,5,6,4,1,2] => [4,6,1,5,3,2] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [3,5,6,1,4,2] => [6,1,4,5,3,2] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [3,5,1,4,2,6] => [4,5,1,3,2,6] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,6,2,4,1,5] => [4,6,2,3,1,5] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,6,2,1,5,4] => [5,6,2,3,1,4] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [4,2,5,6,1,3] => [2,6,1,5,4,3] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [4,2,5,1,3,6] => [2,5,1,4,3,6] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [4,2,6,3,1,5] => [2,6,3,4,1,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [4,2,6,1,5,3] => [2,5,6,1,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,5,3,6,1,2] => [3,6,1,5,4,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,5,3,1,2,6] => [3,5,1,4,2,6] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,6,3,1,2] => [6,3,5,1,4,2] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => [6,1,5,2,4,3] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,5,1,3,2,6] => [5,1,3,4,2,6] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [4,6,3,2,1,5] => [3,6,2,4,1,5] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [4,6,3,1,5,2] => [3,5,6,1,4,2] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [4,6,1,3,5,2] => [5,6,1,3,4,2] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [4,1,3,2,5,6] => [3,4,1,2,5,6] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [3,4,5,6,7,1,2] => [7,1,6,5,4,3,2] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [3,4,5,6,1,2,7] => [6,1,5,4,3,2,7] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [3,4,5,7,2,1,6] => [7,2,5,4,3,1,6] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => [3,4,5,7,1,6,2] => [6,7,1,5,4,3,2] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => [3,4,5,1,2,6,7] => [5,1,4,3,2,6,7] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [3,4,6,2,7,1,5] => [7,2,4,3,1,6,5] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [3,4,6,2,1,5,7] => [6,2,4,3,1,5,7] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [3,4,6,7,5,1,2] => [5,7,1,6,4,3,2] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [3,4,6,7,1,5,2] => [7,1,5,6,4,3,2] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,1,2,5,3,7,4] => [3,4,6,1,5,2,7] => [5,6,1,4,3,2,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [3,4,7,2,5,1,6] => [5,7,2,4,3,1,6] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [3,4,7,2,1,6,5] => [6,7,2,4,3,1,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => [3,4,7,1,5,6,2] => [5,6,7,1,4,3,2] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [3,5,2,6,7,1,4] => [7,2,3,1,6,5,4] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [3,5,2,6,1,4,7] => [6,2,3,1,5,4,7] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [3,5,2,7,4,1,6] => [7,4,5,2,3,1,6] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [3,5,2,7,1,6,4] => [6,7,2,3,1,5,4] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [3,5,2,1,4,6,7] => [5,2,3,1,4,6,7] => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [3,5,6,4,7,1,2] => [4,7,1,6,5,3,2] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [3,5,6,4,1,2,7] => [4,6,1,5,3,2,7] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [3,5,6,7,4,1,2] => [7,4,6,1,5,3,2] => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [3,5,6,7,1,2,4] => [7,1,6,3,2,5,4] => ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [3,5,6,1,4,2,7] => [6,1,4,5,3,2,7] => ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [3,5,7,4,2,1,6] => [4,7,2,5,3,1,6] => ? = 6 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [3,5,7,4,1,6,2] => [4,6,7,1,5,3,2] => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [3,5,7,1,4,6,2] => [6,7,1,4,5,3,2] => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [3,5,1,4,2,6,7] => [4,5,1,3,2,6,7] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,6,2,4,7,1,5] => [4,7,2,3,1,6,5] => ? = 6 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [3,6,2,4,1,5,7] => [4,6,2,3,1,5,7] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [3,6,2,7,5,1,4] => [5,7,2,3,1,6,4] => ? = 6 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [3,6,2,7,1,5,4] => [7,2,3,1,5,6,4] => ? = 7 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,6,2,1,5,4,7] => [5,6,2,3,1,4,7] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [3,6,7,4,5,1,2] => [4,5,7,1,6,3,2] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [3,6,7,4,1,5,2] => [4,7,1,5,6,3,2] => ? = 6 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [3,6,7,1,5,4,2] => [5,7,1,4,6,3,2] => ? = 6 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [3,6,1,4,5,2,7] => [4,5,6,1,3,2,7] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,7,2,4,5,1,6] => [4,5,7,2,3,1,6] => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,7,2,4,1,6,5] => [4,6,7,2,3,1,5] => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,7,2,1,5,6,4] => [5,6,7,2,3,1,4] => ? = 5 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [3,7,1,4,5,6,2] => [4,5,6,7,1,3,2] => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => [4,2,5,6,7,1,3] => [2,7,1,6,5,4,3] => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => [4,2,5,6,1,3,7] => [2,6,1,5,4,3,7] => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [4,2,5,7,3,1,6] => [2,7,3,5,4,1,6] => ? = 6 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => [4,2,5,7,1,6,3] => [2,6,7,1,5,4,3] => ? = 5 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [4,2,5,1,3,6,7] => [2,5,1,4,3,6,7] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [4,2,6,3,7,1,5] => [2,7,3,4,1,6,5] => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [4,2,6,3,1,5,7] => [2,6,3,4,1,5,7] => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [4,2,6,7,5,1,3] => [2,5,7,1,6,4,3] => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [4,2,6,7,1,5,3] => [2,7,1,5,6,4,3] => ? = 6 - 1
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000238
(load all 4 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
St000238: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [2,1] => 1
[1,0,1,0]
 => [3,1,2] => 3
[1,1,0,0]
 => [2,3,1] => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => 4
[1,0,1,1,0,0]
 => [3,1,4,2] => 3
[1,1,0,0,1,0]
 => [2,4,1,3] => 3
[1,1,0,1,0,0]
 => [4,3,1,2] => 3
[1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 4
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 4
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 4
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 5
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 3
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 3
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 3
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 5
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 6
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => 5
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 6
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 5
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => 6
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => 5
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 6
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 6
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => 5
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => 5
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => 5
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => ? = 7
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,1,2,5,3,7,4] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 6
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 7
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => ? = 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 5
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => ? = 7
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => ? = 7
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => ? = 6
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ? = 6
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => ? = 6
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 5
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => ? = 6
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 7
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => ? = 5
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => ? = 5
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => ? = 6
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => ? = 6
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => ? = 5
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 6
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => ? = 5
Description
The number of indices that are not small weak excedances. A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$.
Matching statistic: St000240
(load all 5 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
St000240: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [2,1] => 1
[1,0,1,0]
 => [3,1,2] => 3
[1,1,0,0]
 => [2,3,1] => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => 4
[1,0,1,1,0,0]
 => [3,1,4,2] => 3
[1,1,0,0,1,0]
 => [2,4,1,3] => 3
[1,1,0,1,0,0]
 => [4,3,1,2] => 3
[1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 4
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 4
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 4
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 5
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 3
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 3
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 3
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 5
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 6
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => 5
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 6
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 5
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => 6
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => 5
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 6
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 6
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => 5
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => 5
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => 5
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => ? = 7
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,1,2,5,3,7,4] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 6
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 7
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => ? = 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 5
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => ? = 7
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => ? = 7
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => ? = 6
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ? = 6
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => ? = 6
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 5
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => ? = 6
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 7
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => ? = 5
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => ? = 5
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => ? = 6
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => ? = 6
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => ? = 5
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 6
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => ? = 5
Description
The number of indices that are not small excedances. A small excedance is an index $i$ for which $\pi_i = i+1$.
The following 1 statistic also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation.