Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000216
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [2,3,1] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 3
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,4,5,3,1] => [5,2,4,1,3] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,5,4,3,1] => [4,2,5,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [2,5,3,4,1] => [3,2,4,5,1] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,4,2,5,1] => [5,3,1,4,2] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,5,2,4,1] => [4,3,1,5,2] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,3,5,2,1] => [4,5,3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,4,2,1] => [3,5,4,1,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,5,2,3,1] => [5,4,1,2,3] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,1,4,5,2] => [5,1,3,4,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,5,4,2] => [4,1,3,5,2] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,1,5,3,2] => [5,1,4,2,3] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,1,4,3,2] => [4,1,5,2,3] => 3
[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] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,4,1,5,3] => [5,2,1,4,3] => 2
[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] => 3
[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] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [2,4,3,1,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,5,2,1,4] => [5,3,1,2,4] => 4
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,4,2,1,5] => [4,3,1,2,5] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => [3,4,1,2,5] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,1,2,5,3] => [5,1,2,4,3] => 3
Description
The absolute length of a permutation. The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 3
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001207
(load all 2 compositions to match this statistic)
Mapping
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 2
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,4,2] => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,4,1,2] => 3
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 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] => [1,2,3,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,5,3,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,5,3,4] => ? = 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [5,2,1,3,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,5,2,3,4] => ? = 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [1,5,3,2,4] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => ? = 4
[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] => [3,1,5,2,4] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [1,3,5,2,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,5,1,2,4] => ? = 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,2,4] => ? = 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,5,1,4] => ? = 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,5,1,3,4] => ? = 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ? = 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [1,4,2,5,3] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,5,2,1,3] => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => ? = 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => ? = 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,4,5,1,2] => ? = 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [4,1,2,5,3] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,1,4,5,2] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [1,2,4,5,3] => ? = 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [1,2,4,3,5] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,5,2,3] => ? = 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => ? = 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,4,1,3,5] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ? = 3
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [4,1,5,2,3] => ? = 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [3,1,4,2,5] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,5,1,3] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [1,3,4,2,5] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,3,2,4,5] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,5,1,2,3] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,4,1,2,5] => ? = 3
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[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] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,2,3,4,5] => [1,2,3,6,4,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [2,1,3,6,4,5] => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,3,4,5] => [2,1,6,3,4,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5] => [1,3,2,6,4,5] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [2,6,3,1,4,5] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [1,6,3,2,4,5] => ? = 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001394
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 83%
Values
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [1,4,2,3] => 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,4,2,3,6,5] => 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,6,2,3,4,5] => 2
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,6,2,3,4,5] => 2
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,2,5,3,4,6] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,4,2,3,6,5,8,7] => ? = 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [1,4,2,3,8,5,6,7] => ? = 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,6,2,3,8,4,5,7] => ? = 3
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 2
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => [1,8,2,3,4,5,6,7] => 3
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,6,2,3,4,5,8,7] => ? = 2
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => [1,8,2,3,4,5,6,7] => 3
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,2,5,3,4,7,6,8] => ? = 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [1,2,7,3,4,5,6,8] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 3
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,2,7,3,4,5,6,8] => ? = 2
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => [1,4,2,3,6,5,10,7,8,9] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => [1,4,2,3,8,5,10,6,7,9] => ? = 3
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => [1,4,2,3,10,5,6,9,7,8] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,10,9,8,7,6,5] => [1,4,2,3,10,5,6,7,8,9] => ? = 3
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => [1,6,2,3,8,4,5,7,10,9] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => [1,6,2,3,10,4,5,7,8,9] => ? = 4
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => [1,8,2,3,10,4,7,5,6,9] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,10,5,4,7,6,9,8,3] => [1,10,2,3,4,7,5,6,9,8] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => [1,10,2,3,4,9,5,6,7,8] => ? = 4
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,8,7,6,5,4,3,10,9] => [1,8,2,3,10,4,5,6,7,9] => ? = 3
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => [1,10,2,3,4,9,5,6,7,8] => ? = 4
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => [1,10,2,3,4,5,8,6,7,9] => ? = 3
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => [1,6,2,3,4,5,8,7,10,9] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => [1,6,2,3,4,5,10,7,8,9] => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => [1,8,2,3,4,5,10,6,7,9] => ? = 4
[1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [4,3,2,1,10,7,6,9,8,5] => [1,10,2,3,4,5,6,9,7,8] => ? = 3
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => [1,8,2,5,3,4,6,7,10,9] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [6,3,2,5,4,1,10,9,8,7] => [1,10,2,5,3,4,6,7,8,9] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [8,3,2,5,4,7,6,1,10,9] => [1,10,2,5,3,4,7,6,8,9] => ? = 2
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => [1,2,5,3,4,7,6,9,8,10] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => [1,2,5,3,4,9,6,7,8,10] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => [1,10,2,7,3,4,5,6,8,9] => ? = 4
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => [1,2,7,3,4,9,5,6,8,10] => ? = 3
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => [1,2,9,3,4,5,8,6,7,10] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => [1,2,9,3,4,5,6,7,8,10] => ? = 3
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [6,5,4,3,2,1,8,7,10,9] => [1,8,2,3,4,5,6,7,10,9] => ? = 3
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => 4
[1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => [1,10,2,7,3,4,5,6,8,9] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => [1,2,7,3,4,5,6,9,8,10] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => [1,2,9,3,4,5,6,7,8,10] => ? = 3
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => [1,10,2,3,6,4,5,7,8,9] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [10,7,4,3,6,5,2,9,8,1] => [1,2,9,3,6,4,5,7,8,10] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [10,9,4,3,6,5,8,7,2,1] => [1,2,3,6,4,5,8,7,9,10] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [10,9,4,3,8,7,6,5,2,1] => [1,2,3,8,4,5,6,7,9,10] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 4
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => [1,2,9,3,4,5,6,7,8,10] => ? = 3
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => [1,2,3,8,4,5,6,7,9,10] => ? = 2
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => [1,2,3,4,7,5,6,8,9,10] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,4,2,3,6,5,8,7,10,9,12,11] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,12,11,10,9] => [1,4,2,3,6,5,8,7,12,9,10,11] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [2,1,4,3,6,5,10,9,8,7,12,11] => [1,4,2,3,6,5,10,7,12,8,9,11] => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,12,9,8,11,10,7] => [1,4,2,3,6,5,12,7,8,11,9,10] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,12,11,10,9,8,7] => [1,4,2,3,6,5,12,7,8,9,10,11] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [2,1,4,3,8,7,6,5,10,9,12,11] => [1,4,2,3,8,5,10,6,7,9,12,11] => ? = 3
Description
The genus of a permutation. The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation $$ n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ), $$ where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.