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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: 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]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> 8
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> 8
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> 6
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> 8
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 6
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> 7
[6,2,2,2]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 7
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4
[6,4,1,1,1]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> 9
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 10
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 9
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001000
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 6
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 8
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 7
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 8
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 6
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 8
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 5
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 7
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 7
[6,2,2,2]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 7
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 7
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 7
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[6,4,1,1,1]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 9
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 10
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 9
[5,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5
[4,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
[6,4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 5
[6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 10
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[5,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[5,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 8
[5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[6,4,3,2]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[6,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[6,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 7
[6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 9
[5,5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001778
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001778: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001778: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [2,3,1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? = 1
[6]
=> [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]
=> [8,7,1,2,3,4,5,6] => ? = 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [8,7,1,2,6,3,4,5] => ? = 4
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [9,7,1,2,3,4,8,5,6] => ? = 6
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [8,4,1,2,6,7,3,5] => ? = 4
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [9,8,1,2,3,7,4,5,6] => ? = 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,10,1,2,3,4,5,9,6,7] => ? = 8
[5,3]
=> [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]
=> [5,8,1,2,6,7,3,4] => ? = 3
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [9,10,1,2,3,4,8,5,6,7] => ? = 7
[4,2,2]
=> [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]
=> [5,4,1,2,6,7,8,3] => ? = 5
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [9,10,1,2,3,8,4,5,6,7] => ? = 5
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 4
[3,2,2,1]
=> [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]
=> [4,3,1,8,6,7,2,5] => ? = 3
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [11,10,1,2,3,4,5,9,6,7,8] => ? = 8
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [11,10,1,2,3,4,9,5,6,7,8] => ? = 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,1,2,9,7,8,3,6] => ? = 5
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [4,3,1,9,8,7,2,5,6] => ? = 5
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [9,8,1,2,6,7,3,4,5] => ? = 3
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [7,6,1,2,3,4,8,9,10,5] => ? = 6
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [12,11,1,2,3,4,5,10,6,7,8,9] => ? = 8
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [5,8,1,2,6,7,3,9,4] => ? = 4
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [11,10,1,2,3,4,12,5,6,7,8,9] => ? = 6
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [7,8,4,9,1,2,3,5,6] => ? = 5
[6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? = 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [5,4,1,2,6,7,8,9,3] => ? = 5
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? => ? = 5
[5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [10,6,1,2,3,7,8,9,4,5] => ? = 6
[5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? => ? = 7
[4,4,3]
=> [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]
=> [6,3,4,5,1,7,8,2] => ? = 4
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? => ? = 4
[6,4,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [10,9,1,2,6,11,3,4,5,7,8] => ? = 7
[6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? => ? = 7
[5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? => ? = 7
[5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? => ? = 7
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [7,9,6,5,1,2,8,3,4] => ? = 5
[4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? => ? = 3
[4,2,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> ? => ? = 6
[6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? => ? = 4
[6,4,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 9
[6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 10
[6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 9
[5,5,1,1,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [11,10,9,5,1,2,3,4,6,7,8] => ? = 6
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? => ? = 5
[4,3,3,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 6
[6,4,4]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [5,10,1,2,6,7,8,9,3,4] => ? = 5
[6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ?
=> ? => ? = 7
[6,2,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 10
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [9,6,4,5,1,7,8,2,3] => ? = 4
Description
The largest greatest common divisor of an element and its image in a permutation.
Matching statistic: St001488
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Values
[1]
=> 10 => [1,1] => [[1,1],[]]
=> 2 = 1 + 1
[2]
=> 100 => [1,2] => [[2,1],[]]
=> 3 = 2 + 1
[3]
=> 1000 => [1,3] => [[3,1],[]]
=> 3 = 2 + 1
[2,1]
=> 1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> 2 = 1 + 1
[4]
=> 10000 => [1,4] => [[4,1],[]]
=> 3 = 2 + 1
[3,1]
=> 10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 4 = 3 + 1
[2,1,1]
=> 10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> 4 = 3 + 1
[5]
=> 100000 => [1,5] => [[5,1],[]]
=> ? = 2 + 1
[3,1,1]
=> 100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 1 + 1
[6]
=> 1000000 => [1,6] => [[6,1],[]]
=> ? = 2 + 1
[4,2]
=> 100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ? = 3 + 1
[3,1,1,1]
=> 1001110 => [1,2,3,1] => [[4,4,2,1],[3,1]]
=> ? = 4 + 1
[5,1,1]
=> 10000110 => [1,4,2,1] => [[5,5,4,1],[4,3]]
=> ? = 6 + 1
[4,2,1]
=> 1001010 => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ? = 4 + 1
[4,1,1,1]
=> 10001110 => [1,3,3,1] => [[5,5,3,1],[4,2]]
=> ? = 1 + 1
[6,1,1]
=> 100000110 => [1,5,2,1] => [[6,6,5,1],[5,4]]
=> ? = 8 + 1
[5,3]
=> 1001000 => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ? = 3 + 1
[5,1,1,1]
=> 100001110 => [1,4,3,1] => [[6,6,4,1],[5,3]]
=> ? = 7 + 1
[4,2,2]
=> 1001100 => [1,2,2,2] => [[4,3,2,1],[2,1]]
=> ? = 5 + 1
[4,1,1,1,1]
=> 100011110 => [1,3,4,1] => [[6,6,3,1],[5,2]]
=> ? = 5 + 1
[3,3,2]
=> 110100 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ? = 4 + 1
[3,2,2,1]
=> 1011010 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ? = 3 + 1
[6,1,1,1]
=> 1000001110 => [1,5,3,1] => [[7,7,5,1],[6,4]]
=> ? = 8 + 1
[5,1,1,1,1]
=> 1000011110 => [1,4,4,1] => [[7,7,4,1],[6,3]]
=> ? = 1 + 1
[4,2,2,1]
=> 10011010 => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
=> ? = 5 + 1
[3,2,2,1,1]
=> 10110110 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
=> ? = 5 + 1
[6,4]
=> 10010000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ? = 3 + 1
[6,2,2]
=> 100001100 => [1,4,2,2] => [[6,5,4,1],[4,3]]
=> ? = 6 + 1
[6,1,1,1,1]
=> 10000011110 => [1,5,4,1] => [[8,8,5,1],[7,4]]
=> ? = 8 + 1
[5,3,2]
=> 10010100 => [1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
=> ? = 4 + 1
[5,1,1,1,1,1]
=> 10000111110 => [1,4,5,1] => [[8,8,4,1],[7,3]]
=> ? = 6 + 1
[4,4,1,1]
=> 11000110 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
=> ? = 5 + 1
[6,1,1,1,1,1]
=> 100000111110 => [1,5,5,1] => [[9,9,5,1],[8,4]]
=> ? = 1 + 1
[5,3,3]
=> 10011000 => [1,2,2,3] => [[5,3,2,1],[2,1]]
=> ? = 5 + 1
[5,3,2,1]
=> 100101010 => [1,2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,1],[1,1,1,1,1,1]]
=> ? = 5 + 1
[5,2,2,2]
=> 100011100 => [1,3,3,2] => [[6,5,3,1],[4,2]]
=> ? = 6 + 1
[5,2,2,1,1]
=> 1000110110 => [1,3,2,1,2,1] => [[5,5,4,4,3,1],[4,3,3,2]]
=> ? = 7 + 1
[4,4,3]
=> 1101000 => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> ? = 4 + 1
[4,2,2,2,1]
=> 100111010 => [1,2,3,1,1,1] => [[4,4,4,4,2,1],[3,3,3,1]]
=> ? = 4 + 1
[6,4,1,1]
=> 1001000110 => ? => ?
=> ? = 7 + 1
[6,2,2,2]
=> 1000011100 => ? => ?
=> ? = 7 + 1
[5,3,3,1]
=> 100110010 => [1,2,2,2,1,1] => [[4,4,4,3,2,1],[3,3,2,1]]
=> ? = 7 + 1
[5,2,2,2,1]
=> 1000111010 => [1,3,3,1,1,1] => [[5,5,5,5,3,1],[4,4,4,2]]
=> ? = 7 + 1
[4,4,2,2]
=> 11001100 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
=> ? = 5 + 1
[4,3,3,1,1]
=> 101100110 => [1,1,2,2,2,1] => [[4,4,3,2,1,1],[3,2,1]]
=> ? = 3 + 1
[4,2,2,2,1,1]
=> 1001110110 => ? => ?
=> ? = 6 + 1
[6,4,3]
=> 100101000 => [1,2,1,1,1,3] => [[4,2,2,2,2,1],[1,1,1,1]]
=> ? = 4 + 1
[6,4,1,1,1]
=> 10010001110 => ? => ?
=> ? = 9 + 1
[6,2,2,2,1]
=> 10000111010 => ? => ?
=> ? = 10 + 1
[6,2,2,1,1,1]
=> 100001101110 => [1,4,2,1,3,1] => ?
=> ? = 9 + 1
[5,5,1,1,1]
=> 1100001110 => ? => ?
=> ? = 6 + 1
[5,3,3,1,1]
=> 1001100110 => [1,2,2,2,2,1] => [[5,5,4,3,2,1],[4,3,2,1]]
=> ? = 5 + 1
[4,3,3,1,1,1]
=> 1011001110 => ? => ?
=> ? = 6 + 1
[6,4,4]
=> 100110000 => [1,2,2,4] => [[6,3,2,1],[2,1]]
=> ? = 5 + 1
[6,2,2,2,2]
=> 10000111100 => ? => ?
=> ? = 7 + 1
[6,2,2,2,1,1]
=> 100001110110 => [1,4,3,1,2,1] => ?
=> ? = 10 + 1
[5,5,4]
=> 11010000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ? = 4 + 1
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
Matching statistic: St000455
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Values
[1]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 3 - 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
=> ? = 4 - 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 2
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 8 - 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7 - 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 5 - 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 5 - 2
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 8 - 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ([(0,9),(1,9),(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 - 2
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ([(0,4),(1,4),(2,7),(3,5),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 2
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => ([(0,7),(1,4),(2,5),(2,6),(3,5),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 2
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 2
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [6,1,2,3,4,7,8,9,5] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 6 - 2
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,11,6,7,8,9] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 8 - 2
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => ([(0,7),(1,7),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4 - 2
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,10,5,6,7,8,9] => ([(0,10),(1,10),(2,10),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 6 - 2
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8,7,4,1,2,3,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 2
[6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [12,1,2,3,4,5,11,6,7,8,9,10] => ?
=> ? = 1 - 2
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 5 - 2
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [9,1,2,5,6,3,8,4,7] => ?
=> ? = 5 - 2
[5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [9,1,2,3,6,7,8,4,5] => ?
=> ? = 6 - 2
[5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,10,9,4,7,8] => ?
=> ? = 7 - 2
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [9,1,2,5,6,8,3,4,7] => ?
=> ? = 4 - 2
[6,4,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [10,1,2,9,6,3,4,5,7,8] => ([(0,9),(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7 - 2
[6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [10,1,2,3,4,7,8,9,5,6] => ?
=> ? = 7 - 2
[5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [4,1,2,5,6,9,8,3,7] => ?
=> ? = 7 - 2
[5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,3,6,7,9,4,5,8] => ?
=> ? = 7 - 2
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [6,8,5,1,2,7,3,4] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 5 - 2
[4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,1,4,5,9,8,2,6,7] => ?
=> ? = 3 - 2
[4,2,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [9,1,2,5,6,10,3,4,7,8] => ?
=> ? = 6 - 2
[6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [7,1,2,5,6,3,8,9,4] => ?
=> ? = 4 - 2
[6,4,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9 - 2
[6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 10 - 2
[6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9 - 2
[5,5,1,1,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,10,5,1,2,3,4,6,7,8] => ?
=> ? = 6 - 2
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,10,9,3,7,8] => ?
=> ? = 5 - 2
[4,3,3,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 6 - 2
[6,4,4]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [9,1,2,5,6,7,8,3,4] => ([(0,8),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 5 - 2
[6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 7 - 2
[6,2,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 2
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [8,5,4,1,6,7,2,3] => ([(0,3),(0,4),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000847
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000847: Binary words ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000847: Binary words ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 30%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1010 => 3 = 1 + 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 101100 => 4 = 2 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 2 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 110100 => 3 = 1 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 5 = 3 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 5 = 3 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 2 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 1 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 2 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? = 3 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => ? = 4 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 111011100000 => ? = 6 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 4 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? = 1 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 11101111000000 => ? = 8 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 110011100100 => ? = 3 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 111101100000 => ? = 7 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 5 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 111110010000 => ? = 5 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? = 4 + 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]
=> 1101001010 => ? = 3 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 11110111000000 => ? = 8 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 111110100000 => ? = 1 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => ? = 5 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 110110001010 => ? = 5 + 2
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 11001111001000 => ? = 3 + 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 11100111000010 => ? = 6 + 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 11111011000000 => ? = 8 + 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 111001100100 => ? = 4 + 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 11111100100000 => ? = 6 + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => ? = 5 + 2
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 11111101000000 => ? = 1 + 2
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? = 5 + 2
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 110101100100 => ? = 5 + 2
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 111100100010 => ? = 6 + 2
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 110110100010 => ? = 7 + 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 111000110010 => ? = 4 + 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 111010010010 => ? = 4 + 2
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> 10110111001000 => ? = 7 + 2
[6,2,2,2]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 11110011000010 => ? = 7 + 2
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 110100101100 => ? = 7 + 2
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 111010100010 => ? = 7 + 2
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 111100110000 => ? = 5 + 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 110010110100 => ? = 3 + 2
[4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> 11101100010010 => ? = 6 + 2
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> 11100011001010 => ? = 4 + 2
[6,4,1,1,1]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> 10111011001000 => ? = 9 + 2
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> 11101011000010 => ? = 10 + 2
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> 11011101000010 => ? = 9 + 2
[5,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> 10111011100000 => ? = 6 + 2
[5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => ? = 5 + 2
[4,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> 11001100110100 => ? = 6 + 2
[6,4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> 11100011011000 => ? = 5 + 2
[6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 11111001000010 => ? = 7 + 2
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> 11101101000010 => ? = 10 + 2
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> 11100011100100 => ? = 4 + 2
Description
The number of standard Young tableaux whose descent set is the binary word.
A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation.
For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
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