Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001067
(load all 7 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000932
(load all 7 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => ? = 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001223
(load all 7 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[1,2,3,4,6,5,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[1,2,3,4,7,5,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,3,4,7,6,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,5,6,4,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[1,2,3,5,7,4,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,3,5,7,6,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,6,7,4,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,3,6,7,5,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,2,3,7,6,5,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,5,6,3,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[1,2,4,5,7,3,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,4,5,7,6,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,6,7,3,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,2,4,6,7,5,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,2,4,7,6,5,3] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Matching statistic: St001233
(load all 2 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1,0]
 => 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
 => 0
[2,1] => [1,2] => [2] => [1,1,0,0]
 => 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,3,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[2,1,3] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 0
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
 => 2
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[1,3,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[1,4,2,3] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[1,4,3,2] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,3,4,1] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,4,3,1] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,2,4,1] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[3,4,2,1] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,1,3] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,3,1,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
Matching statistic: St001948
(load all 16 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => ? = 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 1
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => 1
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => 2
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [4,2,1,3] => 0
[1,4,2,3] => [4,1,3,2] => 0
[1,4,3,2] => [4,1,2,3] => 1
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,2,4,1] => 0
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 0
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => 2
[3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 2
[4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => 2
[4,3,2,1] => [1,2,3,4] => 3
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [5,4,2,1,3] => 0
[1,2,5,3,4] => [5,4,1,3,2] => 0
[1,2,5,4,3] => [5,4,1,2,3] => 1
[1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [5,3,2,1,4] => 0
[1,3,5,2,4] => [5,3,1,4,2] => 0
[1,3,5,4,2] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [5,2,4,3,1] => 0
[1,4,2,5,3] => [5,2,4,1,3] => 0
[1,4,3,2,5] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [5,2,3,1,4] => 0
[1,4,5,2,3] => [5,2,1,4,3] => 0
[1,4,5,3,2] => [5,2,1,3,4] => 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 0
[1,2,3,4,6,5] => [6,5,4,3,1,2] => ? = 0
[1,2,3,5,4,6] => [6,5,4,2,3,1] => ? = 0
[1,2,3,5,6,4] => [6,5,4,2,1,3] => ? = 0
[1,2,3,6,4,5] => [6,5,4,1,3,2] => ? = 0
[1,2,3,6,5,4] => [6,5,4,1,2,3] => ? = 1
[1,2,4,3,5,6] => [6,5,3,4,2,1] => ? = 0
[1,2,4,3,6,5] => [6,5,3,4,1,2] => ? = 0
[1,2,4,5,3,6] => [6,5,3,2,4,1] => ? = 0
[1,2,4,5,6,3] => [6,5,3,2,1,4] => ? = 0
[1,2,4,6,3,5] => [6,5,3,1,4,2] => ? = 0
[1,2,4,6,5,3] => [6,5,3,1,2,4] => ? = 1
[1,2,5,3,4,6] => [6,5,2,4,3,1] => ? = 0
[1,2,5,3,6,4] => [6,5,2,4,1,3] => ? = 0
[1,2,5,4,3,6] => [6,5,2,3,4,1] => ? = 1
[1,2,5,4,6,3] => [6,5,2,3,1,4] => ? = 0
[1,2,5,6,3,4] => [6,5,2,1,4,3] => ? = 0
[1,2,5,6,4,3] => [6,5,2,1,3,4] => ? = 1
[1,2,6,3,4,5] => [6,5,1,4,3,2] => ? = 0
[1,2,6,3,5,4] => [6,5,1,4,2,3] => ? = 0
[1,2,6,4,3,5] => [6,5,1,3,4,2] => ? = 1
[1,2,6,4,5,3] => [6,5,1,3,2,4] => ? = 0
[1,2,6,5,3,4] => [6,5,1,2,4,3] => ? = 1
[1,2,6,5,4,3] => [6,5,1,2,3,4] => ? = 2
[1,3,2,4,5,6] => [6,4,5,3,2,1] => ? = 0
[1,3,2,4,6,5] => [6,4,5,3,1,2] => ? = 0
[1,3,2,5,4,6] => [6,4,5,2,3,1] => ? = 0
[1,3,2,5,6,4] => [6,4,5,2,1,3] => ? = 0
[1,3,2,6,4,5] => [6,4,5,1,3,2] => ? = 0
[1,3,2,6,5,4] => [6,4,5,1,2,3] => ? = 1
[1,3,4,2,5,6] => [6,4,3,5,2,1] => ? = 0
[1,3,4,2,6,5] => [6,4,3,5,1,2] => ? = 0
[1,3,4,5,2,6] => [6,4,3,2,5,1] => ? = 0
[1,3,4,5,6,2] => [6,4,3,2,1,5] => ? = 0
[1,3,4,6,2,5] => [6,4,3,1,5,2] => ? = 0
[1,3,4,6,5,2] => [6,4,3,1,2,5] => ? = 1
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ? = 0
[1,3,5,2,6,4] => [6,4,2,5,1,3] => ? = 0
[1,3,5,4,2,6] => [6,4,2,3,5,1] => ? = 1
[1,3,5,4,6,2] => [6,4,2,3,1,5] => ? = 0
[1,3,5,6,2,4] => [6,4,2,1,5,3] => ? = 0
[1,3,5,6,4,2] => [6,4,2,1,3,5] => ? = 1
[1,3,6,2,4,5] => [6,4,1,5,3,2] => ? = 0
[1,3,6,2,5,4] => [6,4,1,5,2,3] => ? = 0
[1,3,6,4,2,5] => [6,4,1,3,5,2] => ? = 1
[1,3,6,4,5,2] => [6,4,1,3,2,5] => ? = 0
[1,3,6,5,2,4] => [6,4,1,2,5,3] => ? = 1
[1,3,6,5,4,2] => [6,4,1,2,3,5] => ? = 2
[1,4,2,3,5,6] => [6,3,5,4,2,1] => ? = 0
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St000454
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => [2] => ([],2)
 => 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0
[3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,2,4,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,3,2,4] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,1,3,4] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,3,1,4] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[2,3,4,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,4,1,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[2,4,3,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,1,2,4] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,1,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,4,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[3,4,2,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,2,3] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,1,3] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,3,1,2] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,2,4,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,3,2,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,4,2,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[2,1,3,4,5] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[6,4,3,2,1,5] => [6,4,3,2,1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[6,5,3,2,1,4] => [6,5,3,2,1,4] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[6,5,4,2,1,3] => [6,5,4,2,1,3] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => ([(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)
 => 5
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[7,5,4,3,2,1,6] => [7,5,4,3,2,1,6] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.