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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St001210
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 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: St001231
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 2 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 4 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 5 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 4 = 6 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 4 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3 = 5 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 4 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 3 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 5 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 4 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 4 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 3 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 5 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 4 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 3 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 5 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 3 - 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 3 - 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 2 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 4 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 5 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 4 = 6 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 4 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3 = 5 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 4 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 3 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 5 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 4 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 4 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 3 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 5 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 4 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 3 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 5 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 3 - 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 3 - 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Matching statistic: St000444
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ 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%
Mp00142: Dyck paths —promotion⟶ 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]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7 = 6 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5 = 4 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 6 = 5 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 5 = 4 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 6 = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 4 = 3 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 6 = 5 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001219
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ 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%
Mp00142: Dyck paths —promotion⟶ 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]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3 = 5 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 4 = 6 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 3 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 0 = 2 - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2 = 4 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 3 - 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
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: St001133
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 4
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10)]
=> 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [(1,6),(2,7),(3,8),(4,9),(5,10),(11,12)]
=> ? = 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10)]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
=> ? = 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [(1,5),(2,7),(3,8),(4,9),(6,10),(11,12)]
=> ? = 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10)]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10)]
=> 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10)]
=> 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
=> ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [(1,5),(2,6),(3,8),(4,9),(7,10),(11,12)]
=> ? = 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [(1,4),(2,7),(3,8),(5,9),(6,10),(11,12)]
=> ? = 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10)]
=> 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10)]
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [(1,3),(2,5),(4,6),(7,9),(8,10)]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
=> ? = 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
=> ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [(1,5),(2,6),(3,7),(4,9),(8,10),(11,12)]
=> ? = 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [(1,4),(2,6),(3,8),(5,9),(7,10),(11,12)]
=> ? = 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [(1,3),(2,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12)]
=> ? = 5
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10)]
=> 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,12)]
=> ? = 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
=> ? = 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)]
=> ? = 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [(1,2),(3,5),(4,9),(6,10),(7,11),(8,12)]
=> ? = 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10),(11,12)]
=> ? = 5
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9),(11,12)]
=> [(1,4),(2,6),(3,7),(5,9),(8,10),(11,12)]
=> ? = 4
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
=> [(1,4),(2,5),(3,8),(6,9),(7,10),(11,12)]
=> ? = 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> [(1,3),(2,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,12),(10,11)]
=> [(1,4),(2,6),(3,7),(5,8),(9,11),(10,12)]
=> ? = 4
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [(1,2),(3,6),(4,8),(5,9),(7,11),(10,12)]
=> ? = 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)]
=> ? = 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> [(1,2),(3,6),(4,7),(5,10),(8,11),(9,12)]
=> ? = 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [(1,3),(2,4),(5,8),(6,10),(7,11),(9,12)]
=> ? = 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,10),(7,11),(9,12)]
=> ? = 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10),(11,12)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10),(11,12)]
=> ? = 4
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10),(11,12)]
=> ? = 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [(1,3),(2,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [(1,3),(2,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]
=> [(1,4),(2,5),(3,7),(6,8),(9,11),(10,12)]
=> ? = 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [(1,3),(2,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 3
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]
=> [(1,4),(2,5),(3,6),(7,9),(8,11),(10,12)]
=> ? = 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [(1,2),(3,6),(4,7),(5,9),(8,11),(10,12)]
=> ? = 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [(1,3),(2,4),(5,8),(6,9),(7,11),(10,12)]
=> ? = 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [(1,2),(3,5),(4,8),(6,9),(7,11),(10,12)]
=> ? = 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [(1,3),(2,5),(4,6),(7,10),(8,11),(9,12)]
=> ? = 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [(1,3),(2,4),(5,7),(6,10),(8,11),(9,12)]
=> ? = 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> 2
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10),(11,12)]
=> ? = 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10),(11,12)]
=> ? = 3
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10),(11,12)]
=> ? = 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10),(11,12)]
=> ? = 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 2
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10),(11,12)]
=> ? = 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> [(1,4),(2,5),(3,6),(7,8),(9,11),(10,12)]
=> ? = 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [(1,3),(2,5),(4,7),(6,8),(9,11),(10,12)]
=> ? = 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 2
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> 2
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> 2
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> 2
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> 2
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> 2
Description
The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001134
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001134: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001134: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 4
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10)]
=> 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [(1,6),(2,7),(3,8),(4,9),(5,10),(11,12)]
=> ? = 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10)]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
=> ? = 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [(1,5),(2,7),(3,8),(4,9),(6,10),(11,12)]
=> ? = 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10)]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10)]
=> 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10)]
=> 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
=> ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [(1,5),(2,6),(3,8),(4,9),(7,10),(11,12)]
=> ? = 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [(1,4),(2,7),(3,8),(5,9),(6,10),(11,12)]
=> ? = 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10)]
=> 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10)]
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [(1,3),(2,5),(4,6),(7,9),(8,10)]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
=> ? = 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
=> ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [(1,5),(2,6),(3,7),(4,9),(8,10),(11,12)]
=> ? = 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [(1,4),(2,6),(3,8),(5,9),(7,10),(11,12)]
=> ? = 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [(1,3),(2,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12)]
=> ? = 5
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10)]
=> 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,12)]
=> ? = 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
=> ? = 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)]
=> ? = 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [(1,2),(3,5),(4,9),(6,10),(7,11),(8,12)]
=> ? = 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10),(11,12)]
=> ? = 5
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9),(11,12)]
=> [(1,4),(2,6),(3,7),(5,9),(8,10),(11,12)]
=> ? = 4
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
=> [(1,4),(2,5),(3,8),(6,9),(7,10),(11,12)]
=> ? = 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> [(1,3),(2,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,12),(10,11)]
=> [(1,4),(2,6),(3,7),(5,8),(9,11),(10,12)]
=> ? = 4
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [(1,2),(3,6),(4,8),(5,9),(7,11),(10,12)]
=> ? = 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)]
=> ? = 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> [(1,2),(3,6),(4,7),(5,10),(8,11),(9,12)]
=> ? = 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [(1,3),(2,4),(5,8),(6,10),(7,11),(9,12)]
=> ? = 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,10),(7,11),(9,12)]
=> ? = 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10),(11,12)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10),(11,12)]
=> ? = 4
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10),(11,12)]
=> ? = 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [(1,3),(2,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [(1,3),(2,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]
=> [(1,4),(2,5),(3,7),(6,8),(9,11),(10,12)]
=> ? = 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [(1,3),(2,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 3
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]
=> [(1,4),(2,5),(3,6),(7,9),(8,11),(10,12)]
=> ? = 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [(1,2),(3,6),(4,7),(5,9),(8,11),(10,12)]
=> ? = 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [(1,3),(2,4),(5,8),(6,9),(7,11),(10,12)]
=> ? = 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [(1,2),(3,5),(4,8),(6,9),(7,11),(10,12)]
=> ? = 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [(1,3),(2,5),(4,6),(7,10),(8,11),(9,12)]
=> ? = 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [(1,3),(2,4),(5,7),(6,10),(8,11),(9,12)]
=> ? = 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> 2
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10),(11,12)]
=> ? = 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10),(11,12)]
=> ? = 3
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10),(11,12)]
=> ? = 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10),(11,12)]
=> ? = 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 2
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10),(11,12)]
=> ? = 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> [(1,4),(2,5),(3,6),(7,8),(9,11),(10,12)]
=> ? = 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [(1,3),(2,5),(4,7),(6,8),(9,11),(10,12)]
=> ? = 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 2
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> 2
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> 2
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> 2
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> 2
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> 2
Description
The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St000740
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,4,2,3] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => 4
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,6,4,5] => 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,5,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,3,1,2] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,2,3,4,7,5,6] => 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [5,2,3,6,1,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,1,4,2] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,6,3,4,1,2] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [7,1,3,4,5,6,2] => ? = 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [6,2,3,4,7,1,5] => ? = 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,5,3,6,2,4] => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,4,3] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,6,3,5,4] => 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,6,4,3,2] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,6,2,4,5,3] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,6,3,1,5,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [6,7,3,4,5,1,2] => ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [1,6,3,4,7,2,5] => ? = 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [6,2,3,1,7,5,4] => ? = 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,5,6,3,4] => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,1,3,6,4,2] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,2,6,1,5,3] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,5,6,4,2,3] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,3,1,2] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [6,1,7,4,5,3,2] => ? = 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,1,4,5,2] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [5,7,3,4,1,6,2] => ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [1,2,6,4,7,3,5] => 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,6,3,2,7,1,4] => ? = 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [6,2,1,4,7,5,3] => ? = 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [1,2,3,7,4,6,5] => 5
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [4,2,5,6,1,3] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,5,2,6,4,3] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,5,3,6,1,2] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [6,1,3,7,5,4,2] => ? = 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,4,6,2,5,3] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,1,6,3,5,2] => 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [3,5,6,4,1,2] => 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,6,7,4,3,1,2] => ? = 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,2,4,5,6,3] => ? = 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [4,7,3,1,5,6,2] => ? = 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,2,3,6,7,4,5] => 5
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [5,2,6,3,7,1,4] => ? = 4
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [1,6,3,2,7,5,4] => ? = 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [5,6,2,4,7,1,3] => ? = 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [6,1,3,4,7,5,2] => ? = 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [5,2,3,7,1,6,4] => ? = 4
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,4,5,6,2,3] => 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,3,2] => 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,5,1,6,4,2] => 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [5,6,3,7,4,1,2] => ? = 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [1,2,7,3,5,6,4] => 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [3,4,6,1,5,2] => 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [5,1,7,4,3,6,2] => ? = 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [1,6,7,4,5,2,3] => ? = 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,6,7,3,5,1,2] => ? = 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [3,7,1,4,5,6,2] => ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [5,2,3,6,7,1,4] => ? = 4
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [1,5,6,3,7,2,4] => ? = 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [5,2,6,4,7,1,3] => ? = 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [4,6,2,1,7,5,3] => ? = 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [5,6,3,4,7,1,2] => ? = 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [1,5,3,7,2,6,4] => ? = 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [5,2,1,7,4,6,3] => ? = 3
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [1,2,6,7,5,3,4] => 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,1,6,7,4,3,2] => ? = 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [1,6,2,7,5,4,3] => ? = 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [4,6,3,7,5,1,2] => ? = 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [4,2,7,1,5,6,3] => ? = 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [1,5,7,4,2,6,3] => ? = 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [4,5,7,3,1,6,2] => ? = 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [3,6,7,4,5,1,2] => ? = 2
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [1,5,3,6,7,2,4] => ? = 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [5,2,1,6,7,4,3] => ? = 3
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [1,2,6,3,7,5,4] => 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,5,6,1,7,2,3] => ? = 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [5,1,6,4,7,3,2] => ? = 2
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [1,6,2,4,7,5,3] => ? = 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [4,6,3,1,7,5,2] => ? = 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [1,2,5,7,3,6,4] => 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [4,5,2,7,1,6,3] => ? = 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [5,1,3,7,4,6,2] => ? = 2
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [4,2,6,7,5,1,3] => ? = 3
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [1,5,6,7,4,2,3] => ? = 3
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,5,6,7,1,3,2] => ? = 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [3,6,1,7,5,4,2] => ? = 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,4,7,2,5,6,3] => ? = 3
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,2,5,6,7,3,4] => 4
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000054
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 80%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 4
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,3,2,6,1] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? = 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,3,5,2,6,1] => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,5,4,3,1] => ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => ? = 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => ? = 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,2,6,1] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,2,6,5,1] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,5,6,4,1] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,3,1] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,5,7,4,3,1] => ? = 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,6,4,3,1] => ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => ? = 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => ? = 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ? = 5
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,6,1] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,6,5,4,7,3,1] => ? = 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,4,1] => 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,4,3,1] => ? = 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,6,5,3,1] => ? = 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 5
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => ? = 4
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,3,6,5,2,7,1] => ? = 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [4,5,3,2,7,6,1] => ? = 4
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => ? = 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ? = 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,4,7,6,3,1] => ? = 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,6,7,5,4,1] => ? = 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => ? = 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 4
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [4,3,5,6,2,7,1] => ? = 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [3,5,4,6,2,7,1] => ? = 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => ? = 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,3,5,2,7,6,1] => ? = 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [3,5,4,2,7,6,1] => ? = 3
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,2,6,7,5,1] => ? = 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => ? = 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => ? = 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,5,7,3,1] => ? = 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,4,2,7,6,5,1] => ? = 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,5,7,6,4,1] => ? = 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,6,3,1] => ? = 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,5,4,1] => ? = 2
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,3,5,2,6,7,1] => 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 3
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => ? = 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,4,6,3,7,1] => ? = 2
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => ? = 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => ? = 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => ? = 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [3,4,5,2,7,6,1] => ? = 3
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => 4
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => 3
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => 3
[5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => 3
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => 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: St000193
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 80%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 1 = 2 - 1
[2]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 4 = 5 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 6 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3 = 4 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 3 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 3 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 5 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3 = 4 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3 = 4 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 2 = 3 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
=> ? = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 5 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3 = 4 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> ? = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 5 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 2 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 3 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 2 - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 3 - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 2 - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 - 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 4 - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 2 - 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 4 - 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 - 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 2 - 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 2 - 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 2 - 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 4 - 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 3 - 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 4 - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 2 - 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 3 - 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 2 - 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 1
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 - 1
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 2 - 1
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 4 - 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 2 - 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 3 - 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 2 - 1
Description
The row of the unique '1' in the first column of the alternating sign matrix.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000051The size of the left subtree of a binary tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000738The first entry in the last row of a standard tableau. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001557The number of inversions of the second entry of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001570The minimal number of edges to add to make a graph Hamiltonian. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001937The size of the center of a parking function.
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