Your data matches 81 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => [1,0,1,0]
 => 1
[1,2] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,1] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,3,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[3,1,2] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[3,2,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,5,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,1,2,3,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000395
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[2,3,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[3,1,2] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[3,2,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001018
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[2,3,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[3,1,2] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[3,2,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,3,4,5,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,4,5,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,4,5,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,4,5,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,4,5,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,1,2,3,4] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,2,3,4,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,3,4,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,3,4,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,4,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,4,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,4,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,5,7,4,8,3,2,1] => [1,2,3,4,8,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,4,5,7,3,8,2,1] => [1,2,3,8,4,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[5,4,6,7,3,8,2,1] => [1,2,3,8,4,6,7,5] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,5,7,4,3,2,8,1] => [1,2,8,3,4,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,4,5,7,3,2,8,1] => [1,2,8,3,4,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,5,4,3,7,2,8,1] => [1,2,8,3,7,4,5,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,4,5,7,2,3,8,1] => [1,2,3,8,4,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,5,7,4,3,8,1,2] => [1,2,3,8,4,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[5,6,4,7,3,8,1,2] => [1,2,3,8,4,7,5,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[5,4,6,7,3,8,1,2] => [1,2,3,8,4,6,7,5] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[5,4,6,3,7,8,1,2] => [1,2,3,7,8,4,6,5] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[8,7,6,3,2,4,1,5] => [1,5,2,4,3,6,7,8] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,7,8,3,2,4,1,5] => [1,5,2,4,3,6,7,8] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[7,8,3,4,2,5,1,6] => [1,6,2,5,3,4,7,8] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,5,7,4,3,2,1,8] => [1,8,2,3,4,5,7,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[6,5,4,7,3,2,1,8] => [1,8,2,3,4,7,5,6] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[7,6,4,3,5,2,1,8] => [1,8,2,3,5,4,6,7] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
[7,5,4,3,6,2,1,8] => [1,8,2,3,6,4,5,7] => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 9 + 1
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001645
(load all 6 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1] => ([],1)
 => 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 2
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,4,1,5,3] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 5
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,6,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [3,6,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,5,3,1,6] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,6,1,5,3] => [1,5,2,4,6,3] => [3,6,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[2,4,6,3,1,5] => [1,5,2,4,6,3] => [3,6,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
Description
The pebbling number of a connected graph.
Matching statistic: St001000
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 62%
Values
[1] => [1] => [1]
 => [1,0,1,0]
 => 1
[1,2] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,1] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,3,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[3,1,2] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[3,2,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[4,5,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,1,2,3,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,3,4,5,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,4,5,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,4,5,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,4,5,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,4,5,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,1,2,3,4] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,2,3,4,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,3,4,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,3,4,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,4,1,2,3] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,4,2,3,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,4,3,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 7
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 7
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000455
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[1] => [1] => [1] => ([],1)
 => ? = 1 - 6
[1,2] => [1,2] => [1,2] => ([],2)
 => ? = 2 - 6
[2,1] => [1,2] => [1,2] => ([],2)
 => ? = 2 - 6
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 3 - 6
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 3 - 6
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 3 - 6
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 3 - 6
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 4 - 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 6
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 6
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 6
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 6
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 6
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 6
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 5 - 6
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ? = 6 - 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ? = 5 - 6
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ? = 5 - 6
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [5,1,4,2,3,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [5,1,4,2,3,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 6
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [5,1,6,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 6
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [5,1,6,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 6
[1,4,6,2,5,7,3] => [1,4,6,2,5,7,3] => [4,6,1,2,5,7,3] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,4,6,3,2,5,7] => [1,4,6,2,5,7,3] => [4,6,1,2,5,7,3] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,5,2,4,7,3,6] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,5,2,4,7,6,3] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,5,3,2,4,7,6] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,5,3,6,2,4,7] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,5,6,7,2,4,3] => [1,5,6,7,2,4,3] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,5,6,7,3,2,4] => [1,5,6,7,2,4,3] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,6,7,2,3,5,4] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[1,6,7,3,5,4,2] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[1,6,7,4,2,3,5] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[1,6,7,4,3,5,2] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[1,7,2,4,6,3,5] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,7,2,4,6,5,3] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,7,3,2,4,6,5] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,7,3,5,2,4,6] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,7,5,2,4,6,3] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[1,7,5,3,2,4,6] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,1,6,7,3,5,4] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[2,1,6,7,4,3,5] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[2,3,5,1,6,7,4] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[2,3,5,4,1,6,7] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[2,4,1,5,6,7,3] => [1,5,6,7,2,4,3] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,3,1,5,6,7] => [1,5,6,7,2,4,3] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,6,1,7,3,5] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,6,1,7,5,3] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,6,3,1,7,5] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,6,3,5,1,7] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,6,5,1,7,3] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,6,5,3,1,7] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,7,1,5,3,6] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,7,3,6,1,5] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,7,6,1,5,3] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,4,7,6,3,1,5] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,5,7,1,4,6,3] => [1,4,6,2,5,7,3] => [4,6,1,2,5,7,3] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[2,5,7,3,1,4,6] => [1,4,6,2,5,7,3] => [4,6,1,2,5,7,3] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,1,4,6,2,5,7] => [1,4,6,2,5,7,3] => [4,6,1,2,5,7,3] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,1,5,2,4,7,6] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,1,5,6,7,2,4] => [1,5,6,7,2,4,3] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,1,7,2,4,6,5] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,1,7,5,2,4,6] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,2,4,1,5,6,7] => [1,5,6,7,2,4,3] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,2,4,6,1,7,5] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,2,4,6,5,1,7] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,2,4,7,6,1,5] => [1,5,2,4,7,3,6] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,2,5,7,1,4,6] => [1,4,6,2,5,7,3] => [4,6,1,2,5,7,3] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,5,1,6,7,4,2] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[3,5,1,7,2,4,6] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
[3,5,2,1,6,7,4] => [1,6,7,2,3,5,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 7 - 6
[3,5,2,4,6,1,7] => [1,7,2,4,6,3,5] => [1,7,2,4,3,6,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 7 - 6
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000696
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000696: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1,0]
 => [2,1] => 1
[1,2] => [1,2] => [1,0,1,0]
 => [3,1,2] => 2
[2,1] => [1,2] => [1,0,1,0]
 => [3,1,2] => 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[4,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[4,3,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[1,5,2,4,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[1,5,3,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[2,4,1,5,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[2,4,3,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[3,1,5,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[3,2,4,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
[3,4,5,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[3,4,5,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[4,5,1,2,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[4,5,2,3,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[4,5,3,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[4,5,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,1,2,3,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,2,3,4,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,3,4,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,3,4,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,4,1,2,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,4,2,3,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,4,3,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 5
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 5
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 5
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 5
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,5,3,1,6] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,4,6,1,5,3] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5
[2,4,6,3,1,5] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 5
[2,5,1,4,6,3] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[2,5,3,1,4,6] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 6
Description
The number of cycles in the breakpoint graph of a permutation. The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$. This graph decomposes into alternating cycles, which this statistic counts. The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by $$ \frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}), $$ where $(x)_n=x(x-1)\dots(x-n+1)$.
Matching statistic: St001430
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
St001430: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 2
[2,1] => [1,2] => [1,2] => [1,2] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => 3
[1,5,3,2,4] => [1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => 3
[2,4,3,1,5] => [1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => 3
[3,1,5,2,4] => [1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => 3
[3,2,4,1,5] => [1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => 3
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,-5,2,3,-6,4] => ? = 5
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,-5,2,3,-6,4] => ? = 5
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,-3,5,2,-6,4] => ? = 5
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,-3,5,2,-6,4] => ? = 5
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,-6,5,2,-4,3] => ? = 5
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,-6,5,2,-4,3] => ? = 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [-4,1,2,-5,3,6] => ? = 5
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [1,5,2,4,6,3] => [-4,1,2,6,-5,3] => ? = 5
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [1,5,2,4,6,3] => [-4,1,2,6,-5,3] => ? = 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [-4,1,2,-5,3,6] => ? = 5
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [1,5,6,2,4,3] => [-6,1,4,2,-5,3] => ? = 5
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [1,5,6,2,4,3] => [-6,1,4,2,-5,3] => ? = 5
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [1,6,2,4,5,3] => [4,1,2,5,-6,3] => ? = 5
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [1,6,2,4,5,3] => [4,1,2,5,-6,3] => ? = 5
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [5,1,2,3,-6,4] => ? = 5
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [-4,1,2,-5,3,6] => ? = 5
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [1,5,6,2,4,3] => [-6,1,4,2,-5,3] => ? = 5
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [1,5,6,2,4,3] => [-6,1,4,2,-5,3] => ? = 5
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [-4,1,-6,2,3,5] => ? = 5
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [-4,1,2,-5,3,6] => ? = 5
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [1,6,2,4,5,3] => [4,1,2,5,-6,3] => ? = 5
[2,4,5,3,1,6] => [1,6,2,4,5,3] => [1,6,2,4,5,3] => [4,1,2,5,-6,3] => ? = 5
[2,4,6,1,5,3] => [1,5,2,4,6,3] => [1,5,2,4,6,3] => [-4,1,2,6,-5,3] => ? = 5
[2,4,6,3,1,5] => [1,5,2,4,6,3] => [1,5,2,4,6,3] => [-4,1,2,6,-5,3] => ? = 5
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,-3,5,2,-6,4] => ? = 5
[2,5,1,4,6,3] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,-6,5,2,-4,3] => ? = 5
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[2,5,3,1,4,6] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,-6,5,2,-4,3] => ? = 5
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [5,1,-6,2,3,4] => ? = 5
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
Description
The number of positive entries in a signed permutation.
Matching statistic: St000441
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1,0]
 => [2,1] => 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[2,1] => [1,2] => [1,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 2 = 3 - 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 2 = 3 - 1
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[4,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[2,4,1,5,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
[2,4,3,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
[3,1,5,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
[3,2,4,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,3,4,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,3,4,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,4,1,2,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,4,2,3,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 5 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 5 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 5 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 5 - 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5 - 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5 - 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5 - 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5 - 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5 - 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5 - 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5 - 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5 - 1
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5 - 1
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 5 - 1
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 5 - 1
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,5,3,1,6] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,4,6,1,5,3] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5 - 1
[2,4,6,3,1,5] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 5 - 1
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ? = 5 - 1
[2,5,1,4,6,3] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5 - 1
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[2,5,3,1,4,6] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 5 - 1
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 5 - 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 5 = 6 - 1
Description
The number of successions of a permutation. A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000189
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => ([],1)
 => 1
[1,2] => [1,2] => ([(0,1)],2)
 => 2
[2,1] => [1,2] => ([(0,1)],2)
 => 2
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3
[3,1,5,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3
[3,2,4,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 5
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 5
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
 => ? = 5
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
 => ? = 5
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
 => ? = 5
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
 => ? = 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,21),(2,8),(2,16),(2,17),(2,18),(2,21),(3,9),(3,10),(3,17),(3,18),(4,12),(4,13),(4,17),(4,21),(5,11),(5,13),(5,16),(5,18),(6,8),(6,10),(6,11),(6,12),(6,21),(8,15),(8,19),(8,20),(8,25),(9,24),(10,19),(10,20),(10,24),(11,14),(11,20),(11,25),(12,14),(12,15),(12,19),(13,14),(13,25),(14,23),(15,22),(15,23),(16,24),(16,25),(17,19),(17,24),(17,25),(18,20),(18,24),(18,25),(19,22),(19,23),(20,22),(20,23),(21,15),(21,24),(21,25),(22,7),(23,7),(24,22),(25,22),(25,23)],26)
 => ? = 5
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,21),(2,8),(2,16),(2,17),(2,18),(2,21),(3,9),(3,10),(3,17),(3,18),(4,12),(4,13),(4,17),(4,21),(5,11),(5,13),(5,16),(5,18),(6,8),(6,10),(6,11),(6,12),(6,21),(8,15),(8,19),(8,20),(8,25),(9,24),(10,19),(10,20),(10,24),(11,14),(11,20),(11,25),(12,14),(12,15),(12,19),(13,14),(13,25),(14,23),(15,22),(15,23),(16,24),(16,25),(17,19),(17,24),(17,25),(18,20),(18,24),(18,25),(19,22),(19,23),(20,22),(20,23),(21,15),(21,24),(21,25),(22,7),(23,7),(24,22),(25,22),(25,23)],26)
 => ? = 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
 => ? = 5
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
 => ? = 5
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 5
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 5
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 5
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 5
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 5
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 5
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 5
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
 => ? = 5
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
 => ? = 5
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,3,6,1,5] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 5
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,21),(2,8),(2,16),(2,17),(2,18),(2,21),(3,9),(3,10),(3,17),(3,18),(4,12),(4,13),(4,17),(4,21),(5,11),(5,13),(5,16),(5,18),(6,8),(6,10),(6,11),(6,12),(6,21),(8,15),(8,19),(8,20),(8,25),(9,24),(10,19),(10,20),(10,24),(11,14),(11,20),(11,25),(12,14),(12,15),(12,19),(13,14),(13,25),(14,23),(15,22),(15,23),(16,24),(16,25),(17,19),(17,24),(17,25),(18,20),(18,24),(18,25),(19,22),(19,23),(20,22),(20,23),(21,15),(21,24),(21,25),(22,7),(23,7),(24,22),(25,22),(25,23)],26)
 => ? = 5
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,21),(2,8),(2,16),(2,17),(2,18),(2,21),(3,9),(3,10),(3,17),(3,18),(4,12),(4,13),(4,17),(4,21),(5,11),(5,13),(5,16),(5,18),(6,8),(6,10),(6,11),(6,12),(6,21),(8,15),(8,19),(8,20),(8,25),(9,24),(10,19),(10,20),(10,24),(11,14),(11,20),(11,25),(12,14),(12,15),(12,19),(13,14),(13,25),(14,23),(15,22),(15,23),(16,24),(16,25),(17,19),(17,24),(17,25),(18,20),(18,24),(18,25),(19,22),(19,23),(20,22),(20,23),(21,15),(21,24),(21,25),(22,7),(23,7),(24,22),(25,22),(25,23)],26)
 => ? = 5
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,3,4,5,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
Description
The number of elements in the poset.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000070The number of antichains in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001330The hat guessing number of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001820The size of the image of the pop stack sorting operator. St000454The largest eigenvalue of a graph if it is integral. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001616The number of neutral elements in a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001863The number of weak excedances of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000797The stat`` of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001889The size of the connectivity set of a signed permutation. St000062The length of the longest increasing subsequence of the permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000327The number of cover relations in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001637The number of (upper) dissectors of a poset. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001684The reduced word complexity of a permutation. St001434The number of negative sum pairs of a signed permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation.