Your data matches 126 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001231
(load all 2 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => 0
[1,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Matching statistic: St001234
(load all 2 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => 0
[1,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
Description
The number of indecomposable three dimensional modules with projective dimension one. It return zero when there are no such modules.
Matching statistic: St001210
(load all 2 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => 2 = 0 + 2
[1,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 1 + 2
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 0 + 2
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 0 + 2
[1,1,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 1 + 2
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 1 + 2
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 1 + 2
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 2 + 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 2 + 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 2 + 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 2 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 1 + 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 1 + 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 1 + 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 1 + 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 1 + 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 1 + 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
Description
Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001219
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001219: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
Description
Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive.
Matching statistic: St000444
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 0 + 3
[1,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 1 + 3
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 0 + 3
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 0 + 3
[1,1,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 2 + 3
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 1 + 3
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 1 + 3
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 1 + 3
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 0 + 3
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 0 + 3
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 0 + 3
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 0 + 3
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 0 + 3
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 0 + 3
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 0 + 3
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 0 + 3
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 0 + 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 3 + 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 2 + 3
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 2 + 3
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 2 + 3
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 2 + 3
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5 = 2 + 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5 = 2 + 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5 = 2 + 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5 = 2 + 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5 = 2 + 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5 = 2 + 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5 = 2 + 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5 = 2 + 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 1 + 3
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 1 + 3
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 1 + 3
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 1 + 3
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 1 + 3
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 1 + 3
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 1 + 3
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001947
(load all 3 compositions to match this statistic)
Mapping
Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
Mp00301: Parking functions —leading to repeats⟶ Parking functions
St001947: Parking functions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,1] => [1,1] => [1,1] => 1
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,1,1] => [1,1,1] => [1,1,1] => 2
[1,1,2] => [1,1,2] => [1,1,2] => 1
[1,2,1] => [1,1,2] => [1,1,2] => 1
[2,1,1] => [1,1,2] => [1,1,2] => 1
[1,1,3] => [1,1,3] => [1,1,3] => 1
[1,3,1] => [1,1,3] => [1,1,3] => 1
[3,1,1] => [1,1,3] => [1,1,3] => 1
[1,2,2] => [1,2,2] => [1,2,3] => 0
[2,1,2] => [1,2,2] => [1,2,3] => 0
[2,2,1] => [1,2,2] => [1,2,3] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => 0
[1,1,1,1] => [1,1,1,1] => [1,1,1,1] => 3
[1,1,1,2] => [1,1,1,2] => [1,1,1,2] => 2
[1,1,2,1] => [1,1,1,2] => [1,1,1,2] => 2
[1,2,1,1] => [1,1,1,2] => [1,1,1,2] => 2
[2,1,1,1] => [1,1,1,2] => [1,1,1,2] => 2
[1,1,1,3] => [1,1,1,3] => [1,1,1,3] => 2
[1,1,3,1] => [1,1,1,3] => [1,1,1,3] => 2
[1,3,1,1] => [1,1,1,3] => [1,1,1,3] => 2
[3,1,1,1] => [1,1,1,3] => [1,1,1,3] => 2
[1,1,1,4] => [1,1,1,4] => [1,1,1,4] => 2
[1,1,4,1] => [1,1,1,4] => [1,1,1,4] => 2
[1,4,1,1] => [1,1,1,4] => [1,1,1,4] => 2
[4,1,1,1] => [1,1,1,4] => [1,1,1,4] => 2
[1,1,2,2] => [1,1,2,2] => [1,1,2,3] => 1
[1,2,1,2] => [1,1,2,2] => [1,1,2,3] => 1
[1,2,2,1] => [1,1,2,2] => [1,1,2,3] => 1
[2,1,1,2] => [1,1,2,2] => [1,1,2,3] => 1
[2,1,2,1] => [1,1,2,2] => [1,1,2,3] => 1
[2,2,1,1] => [1,1,2,2] => [1,1,2,3] => 1
[1,1,2,3] => [1,1,2,3] => [1,1,2,4] => 1
[1,1,3,2] => [1,1,2,3] => [1,1,2,4] => 1
[1,2,1,3] => [1,1,2,3] => [1,1,2,4] => 1
[1,2,3,1] => [1,1,2,3] => [1,1,2,4] => 1
[1,3,1,2] => [1,1,2,3] => [1,1,2,4] => 1
[1,3,2,1] => [1,1,2,3] => [1,1,2,4] => 1
[2,1,1,3] => [1,1,2,3] => [1,1,2,4] => 1
[2,1,3,1] => [1,1,2,3] => [1,1,2,4] => 1
[2,3,1,1] => [1,1,2,3] => [1,1,2,4] => 1
[3,1,1,2] => [1,1,2,3] => [1,1,2,4] => 1
[3,1,2,1] => [1,1,2,3] => [1,1,2,4] => 1
[1,2,2,3,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,2,5,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,2,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,5,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,5,2,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,5,3,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,2,2,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,2,5,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,5,2,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,5,2,2,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,5,2,3,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,5,3,2,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,2,3,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,2,5,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3,2,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3,5,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,5,2,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,5,3,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1,3,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1,5,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,3,1,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,3,5,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,5,1,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,5,3,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1,2,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1,5,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,2,1,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,2,5,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,5,1,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,5,2,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,5,1,2,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,5,1,3,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,5,2,1,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,5,2,3,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,5,3,1,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,5,3,2,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,2,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,5,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,5,2,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,2,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,5,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,2,1,5] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,2,5,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,5,1,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,5,2,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,5,1,2,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,5,2,1,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,5,2,2,1] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[5,1,2,2,3] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[5,1,2,3,2] => [1,2,2,3,5] => [1,2,3,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
Description
The number of ties in a parking function. This is the number of indices $i$ such that $p_i=p_{i+1}$.
Matching statistic: St001525
(load all 4 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 89%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
 => [1,0]
 => []
 => ? = 0
[1,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {0,0}
[2,1] => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {0,0}
[1,1,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,2,1] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[3,2,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,3,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,4,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,4,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,5,3,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,5,4,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,2,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,2,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,4,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,4,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,2,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,4,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,2,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,2,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,5,2,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001940
(load all 3 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 89%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
 => [1,0]
 => []
 => ? = 0
[1,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {0,0}
[2,1] => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {0,0}
[1,1,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,2,1] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,2}
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,3,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,4,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,4,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,5,3,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,5,4,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,2,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,2,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,4,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,4,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,2,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,4,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,2,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,2,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,5,2,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001124
(load all 12 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 84%●distinct values known / distinct values provided: 80%
Values
[1] => [1,0]
 => [1,0]
 => []
 => ? = 0
[1,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1]
 => ? ∊ {0,0,1}
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {0,0,1}
[2,1] => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {0,0,1}
[1,1,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1
[1,1,2] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,2,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2}
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[3,2,1,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,2,3,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,3,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,3,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,1,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,3,1] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,2,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,3,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,1,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,3,1] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,3,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,3,2,1] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,4,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000506
(load all 4 compositions to match this statistic)
Mapping
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 82%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [[1],[]]
 => []
 => ? = 0
[1,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1}
[1,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1}
[2,1] => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,1}
[1,1,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[1,1,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[1,2,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[2,1,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,1,3] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[3,1,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[2,1,2] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[2,2,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[1,3,2] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[2,1,3] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[2,3,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[3,1,2] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[3,2,1] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,2,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,3] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[3,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,4] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,4,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,4,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,2,2,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[2,2,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,3,2] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,1,3] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,2,3,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,3,2,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,1,3,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[2,3,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[3,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[3,1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[3,2,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,1,2,4] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,4,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,2,1,4] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,2,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,1,4] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[2,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,4,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[4,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[4,1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[4,2,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,1,3,3] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,3,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,3,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[3,1,3,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[3,3,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,1,3,4] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,1,4,3] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,1,4] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,1,3] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,3,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[3,1,1,4] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[3,4,1,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[4,1,1,3] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[4,3,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,2,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,2,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,2,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,2,2,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,2] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,1,3,2] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[2,2,1,3] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,2,3,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,3,2,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,2,4] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,2,4,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,2,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,3,2] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
The following 116 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001937The size of the center of a parking function. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000115The single entry in the last row. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000137The Grundy value of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000177The number of free tiles in the pattern. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001651The Frankl number of a lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St000445The number of rises of length 1 of a Dyck path. St000441The number of successions of a permutation. St000352The Elizalde-Pak rank of a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000904The maximal number of repetitions of an integer composition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000674The number of hills of a Dyck path. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001487The number of inner corners of a skew partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000090The variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000654The first descent of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition.