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Your data matches 26 different statistics following compositions of up to 3 maps.
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Matching statistic: St001231
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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)
(load all 2 compositions to match this statistic)
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%
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)
(load all 2 compositions to match this statistic)
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%
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
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%
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
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%
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: St000352
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 60%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[2,4,1,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[4,1,1,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[4,1,2,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[4,2,1,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2 = 1 + 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2 = 1 + 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2 = 1 + 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2 = 1 + 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2 = 1 + 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2 = 1 + 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000054
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 60%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 2 = 0 + 2
[1,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 3 = 1 + 2
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 0 + 2
[2,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 0 + 2
[1,1,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 4 = 2 + 2
[1,1,2] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 3 = 1 + 2
[1,2,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 3 = 1 + 2
[2,1,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 3 = 1 + 2
[1,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 3 = 1 + 2
[1,3,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 3 = 1 + 2
[3,1,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 3 = 1 + 2
[1,2,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2 = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2 = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[1,3,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[2,1,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[2,3,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[3,1,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[3,2,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 3 + 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 + 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 1 + 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1 + 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 1 + 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 1 + 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 1 + 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 1 + 2
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 1 + 2
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 1 + 2
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 1 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St001355
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 80%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 80%
Values
[1] => [1,0]
=> [1,0]
=> 10 => 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 4 + 1
[1,1,1,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 3 + 1
[1,1,1,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 3 + 1
[1,1,2,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 3 + 1
[1,2,1,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 3 + 1
[2,1,1,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 3 + 1
[1,1,1,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 3 + 1
[1,1,1,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 3 + 1
[1,1,3,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 3 + 1
[1,3,1,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 3 + 1
[3,1,1,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 3 + 1
[1,1,1,1,4] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 3 + 1
[1,1,1,4,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 3 + 1
[1,1,4,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 3 + 1
[1,4,1,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 3 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 3 + 1
[1,1,1,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 3 + 1
[1,1,1,5,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 3 + 1
[1,1,5,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 3 + 1
[1,5,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 3 + 1
[5,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 3 + 1
[1,1,1,2,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[1,1,2,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[1,1,2,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[1,2,1,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[1,2,1,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[1,2,2,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[2,1,1,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[2,1,1,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[2,1,2,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[2,2,1,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 + 1
[1,1,1,2,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,1,1,3,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,1,2,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,1,2,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,1,3,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,1,3,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,2,1,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,2,1,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,2,3,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,3,1,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,3,1,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[1,3,2,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[2,1,1,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[2,1,1,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[2,1,3,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[2,3,1,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[3,1,1,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[3,1,1,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
[3,1,2,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2 + 1
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.
Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Matching statistic: St001462
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 80%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 80%
Values
[1] => [1,0]
=> [1,0]
=> [[1],[2]]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4 + 1
[1,1,1,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 3 + 1
[1,1,1,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 3 + 1
[1,1,2,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 3 + 1
[1,2,1,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 3 + 1
[2,1,1,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 3 + 1
[1,1,1,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> ? = 3 + 1
[1,1,1,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> ? = 3 + 1
[1,1,3,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> ? = 3 + 1
[1,3,1,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> ? = 3 + 1
[3,1,1,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> ? = 3 + 1
[1,1,1,1,4] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> ? = 3 + 1
[1,1,1,4,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> ? = 3 + 1
[1,1,4,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> ? = 3 + 1
[1,4,1,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> ? = 3 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> ? = 3 + 1
[1,1,1,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 3 + 1
[1,1,1,5,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 3 + 1
[1,1,5,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 3 + 1
[1,5,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 3 + 1
[5,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 3 + 1
[1,1,1,2,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[1,1,2,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[1,1,2,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[1,2,1,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[1,2,1,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[1,2,2,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[2,1,1,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[2,1,1,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[2,2,1,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 + 1
[1,1,1,2,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,1,1,3,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,1,2,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,1,2,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,1,3,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,1,3,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,2,1,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,2,1,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,2,3,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,3,1,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,3,1,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[1,3,2,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[2,1,1,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[2,1,1,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[2,1,3,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[2,3,1,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[3,1,1,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[3,1,1,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
[3,1,2,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 2 + 1
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St001553
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 80%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 80%
Values
[1] => [1,0]
=> [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[1,1,1,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,1,2,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,2,1,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[2,1,1,1,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,1,1,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,1,3,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,3,1,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[3,1,1,1,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,1,1,1,4] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,4,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,4,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,4,1,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,5,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,5,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,5,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[5,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,2,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,2,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,2,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,1,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,1,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,2,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[2,1,1,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[2,1,1,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[2,2,1,1,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,1,2,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,1,3,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,2,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,2,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,3,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,3,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,1,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,1,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,3,1,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,3,1,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,3,2,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,1,1,1,3] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,1,1,3,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,1,3,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,3,1,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[3,1,1,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[3,1,1,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[3,1,2,1,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000674The number of hills of a Dyck path. 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. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000492The rob 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. St000654The first descent of a permutation. 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.
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