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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,4,1,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[3,4,1,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[4,3,1,2] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,3,5,2,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,4,5,2,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,5,2,3,4] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,5,3,2,4] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,5,4,2,3] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,1,3,4,5] => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,4,3,5] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,1,5,3,4] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,3,1,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,3,4,1,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,3,4,5,1] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,3,5,1,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,4,1,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,4,3,1,5] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,4,5,1,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,5,1,3,4] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,5,3,1,4] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001227
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 86%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,4,1,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[3,4,1,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[4,3,1,2] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,3,5,2,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,4,5,2,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,5,2,3,4] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,5,3,2,4] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,5,4,2,3] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,1,3,4,5] => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,4,3,5] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,1,5,3,4] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,3,1,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,3,4,1,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,3,4,5,1] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,3,5,1,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,4,1,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,4,3,1,5] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,4,5,1,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,5,1,3,4] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,5,3,1,4] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,2,3,4,5,6,7] => [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,5,7,6] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,2,3,4,6,5,7] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,3,4,6,7,5] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,2,3,4,7,5,6] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,3,5,4,6,7] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,5,6,4,7] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,3,5,6,7,4] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,2,3,5,7,4,6] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,3,6,4,5,7] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,6,5,4,7] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,3,6,7,4,5] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,3,7,4,5,6] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,7,5,4,6] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,3,7,6,4,5] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,4,3,5,6,7] => [3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,2,4,3,6,5,7] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,4,3,7,5,6] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,4,5,3,6,7] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,5,6,3,7] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,4,5,6,7,3] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,2,4,5,7,3,6] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,4,6,3,5,7] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,6,5,3,7] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,4,6,7,3,5] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,4,7,3,5,6] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,7,5,3,6] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,4,7,6,3,5] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,5,3,4,6,7] => [3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,2,5,3,6,4,7] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,5,3,7,4,6] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,5,4,3,6,7] => [3,1,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,2,5,4,6,3,7] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,5,4,7,3,6] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,5,6,3,4,7] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,5,6,4,3,7] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,5,6,7,3,4] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,2,5,7,3,4,6] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,5,7,4,3,6] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,5,7,6,3,4] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,6,3,4,5,7] => [3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,2,6,3,5,4,7] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,6,3,7,4,5] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,6,4,3,5,7] => [3,1,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,2,6,4,5,3,7] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,6,4,7,3,5] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,6,5,3,4,7] => [3,1,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,2,6,5,7,3,4] => [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,6,7,3,4,5] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,6,7,4,3,5] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001772
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 71%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[3,1,2] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,2,4] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 2
[1,3,4,2] => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 3
[1,4,2,3] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,1,3,4] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[2,3,1,4] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 2
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[3,2,1,4] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[3,4,1,2] => [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 2
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => [3,4,2,1] => 1
[4,2,1,3] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 3
[4,3,1,2] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 3
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,2,4,3,5] => [5,4,2,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 3
[1,2,4,5,3] => [5,4,2,1,3] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 4
[1,2,5,3,4] => [5,4,1,3,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[1,3,2,4,5] => [5,3,4,2,1] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 2
[1,3,4,2,5] => [5,3,2,4,1] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 3
[1,3,4,5,2] => [5,3,2,1,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 4
[1,3,5,2,4] => [5,3,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 3
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 4
[1,4,5,2,3] => [5,2,1,4,3] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 3
[1,5,2,3,4] => [5,1,4,3,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[1,5,3,2,4] => [5,1,3,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 4
[1,5,4,2,3] => [5,1,2,4,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 4
[2,1,3,4,5] => [4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[2,1,4,3,5] => [4,5,2,3,1] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 3
[2,1,5,3,4] => [4,5,1,3,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 3
[2,3,1,4,5] => [4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 2
[2,3,4,1,5] => [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 3
[2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 4
[2,3,5,1,4] => [4,3,1,5,2] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 3
[2,4,1,3,5] => [4,2,5,3,1] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[2,4,3,1,5] => [4,2,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 4
[2,4,5,1,3] => [4,2,1,5,3] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 3
[2,5,1,3,4] => [4,1,5,3,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[2,5,3,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 4
[2,5,4,1,3] => [4,1,2,5,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 4
[3,1,2,4,5] => [3,5,4,2,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[3,1,4,2,5] => [3,5,2,4,1] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 3
[3,1,5,2,4] => [3,5,1,4,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3
[3,2,1,4,5] => [3,4,5,2,1] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 4
[3,2,4,1,5] => [3,4,2,5,1] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 3
[3,2,5,1,4] => [3,4,1,5,2] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 3
[3,4,1,2,5] => [3,2,5,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[3,4,2,1,5] => [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 4
[3,4,5,1,2] => [3,2,1,5,4] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 3
[3,5,1,2,4] => [3,1,5,4,2] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[3,5,2,1,4] => [3,1,4,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 4
[3,5,4,1,2] => [3,1,2,5,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 4
[4,1,2,3,5] => [2,5,4,3,1] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
[4,1,3,2,5] => [2,5,3,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 3
[4,1,5,2,3] => [2,5,1,4,3] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 3
[4,2,1,3,5] => [2,4,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 4
[4,2,3,1,5] => [2,4,3,5,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 3
[4,2,5,1,3] => [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3
[4,3,1,2,5] => [2,3,5,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 4
[4,3,5,1,2] => [2,3,1,5,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3
[4,5,1,2,3] => [2,1,5,4,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[4,5,2,1,3] => [2,1,4,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 4
[4,5,3,1,2] => [2,1,3,5,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 4
Description
The number of occurrences of the signed pattern 12 in a signed permutation.
This is the number of pairs 1≤i<j≤n such that 0<π(i)<π(j).
Matching statistic: St001931
Mp00305: Permutations —parking function⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
St001931: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Mp00319: Parking functions —to composition⟶ Integer compositions
St001931: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 1
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 3
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 3
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 3
[1,3,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 4
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 3
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 2
[1,5,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 3
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 3
[2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[2,3,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 3
[2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 4
[2,4,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 3
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[2,5,3,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 4
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 4
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[3,1,4,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[3,2,4,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 3
[3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 3
[3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[3,4,2,1,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 4
[3,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 3
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2
[3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 4
[3,5,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 4
[4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 3
[4,1,5,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 3
[4,2,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 4
[4,2,3,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 3
[4,2,5,1,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3
[4,3,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 4
[4,3,5,1,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 3
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[4,5,2,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 4
Description
The weak major index of an integer composition regarded as a word.
This is the sum of the positions of the weak descents, regarding the composition as a word. That is, for a composition c=(c1,…,cn),
∑1≤i<nci≥ci+1i.
Matching statistic: St000136
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000136: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000136: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 2
[3,1,2] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,2,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 2
[1,3,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 3
[1,4,2,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 2
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 3
[2,4,1,3] => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 3
[3,4,1,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[4,1,2,3] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[4,2,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 3
[4,3,1,2] => [1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 4
[1,2,4,3,5] => [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 3
[1,2,4,5,3] => [2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 4
[1,2,5,3,4] => [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 3
[1,3,2,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 2
[1,3,4,2,5] => [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 3
[1,3,4,5,2] => [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 4
[1,3,5,2,4] => [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 3
[1,4,2,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[1,4,3,2,5] => [3,4,2,1,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 4
[1,4,5,2,3] => [2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 3
[1,5,2,3,4] => [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[1,5,3,2,4] => [3,4,1,5,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 4
[1,5,4,2,3] => [3,1,5,4,2] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[2,1,4,3,5] => [3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 3
[2,1,5,3,4] => [3,2,1,5,4] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 3
[2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 2
[2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 3
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 4
[2,3,5,1,4] => [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[2,4,1,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 4
[2,4,5,1,3] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 3
[2,5,1,3,4] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[2,5,3,1,4] => [4,2,1,5,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 4
[2,5,4,1,3] => [1,4,5,3,2] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 4
[3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[3,1,4,2,5] => [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 3
[3,1,5,2,4] => [3,1,4,5,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 4
[3,2,4,1,5] => [4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 3
[3,2,5,1,4] => [1,4,3,5,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3
[3,4,1,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[3,4,2,1,5] => [4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 4
[3,4,5,1,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 3
[3,5,1,2,4] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[3,5,2,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 4
[3,5,4,1,2] => [1,5,3,4,2] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 4
[4,1,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
[4,1,3,2,5] => [2,4,3,1,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 3
[4,1,5,2,3] => [3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3
[4,2,1,3,5] => [3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 4
[4,2,3,1,5] => [4,1,3,2,5] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 3
[4,2,5,1,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 3
[4,3,1,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 4
[4,3,5,1,2] => [1,5,3,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3
[4,5,1,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[4,5,2,1,3] => [4,1,5,2,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 4
Description
The dinv of a parking function.
Matching statistic: St000194
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000194: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000194: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 2
[3,1,2] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,2,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 2
[1,3,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 3
[1,4,2,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 2
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 3
[2,4,1,3] => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 3
[3,4,1,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[4,1,2,3] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[4,2,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 3
[4,3,1,2] => [1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 4
[1,2,4,3,5] => [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 3
[1,2,4,5,3] => [2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 4
[1,2,5,3,4] => [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 3
[1,3,2,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 2
[1,3,4,2,5] => [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 3
[1,3,4,5,2] => [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 4
[1,3,5,2,4] => [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 3
[1,4,2,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[1,4,3,2,5] => [3,4,2,1,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 4
[1,4,5,2,3] => [2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 3
[1,5,2,3,4] => [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[1,5,3,2,4] => [3,4,1,5,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 4
[1,5,4,2,3] => [3,1,5,4,2] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[2,1,4,3,5] => [3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 3
[2,1,5,3,4] => [3,2,1,5,4] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 3
[2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 2
[2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 3
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 4
[2,3,5,1,4] => [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[2,4,1,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 4
[2,4,5,1,3] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 3
[2,5,1,3,4] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[2,5,3,1,4] => [4,2,1,5,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 4
[2,5,4,1,3] => [1,4,5,3,2] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 4
[3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[3,1,4,2,5] => [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 3
[3,1,5,2,4] => [3,1,4,5,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 4
[3,2,4,1,5] => [4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 3
[3,2,5,1,4] => [1,4,3,5,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3
[3,4,1,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[3,4,2,1,5] => [4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 4
[3,4,5,1,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 3
[3,5,1,2,4] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[3,5,2,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 4
[3,5,4,1,2] => [1,5,3,4,2] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 4
[4,1,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
[4,1,3,2,5] => [2,4,3,1,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 3
[4,1,5,2,3] => [3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3
[4,2,1,3,5] => [3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 4
[4,2,3,1,5] => [4,1,3,2,5] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 3
[4,2,5,1,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 3
[4,3,1,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 4
[4,3,5,1,2] => [1,5,3,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3
[4,5,1,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[4,5,2,1,3] => [4,1,5,2,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 4
Description
The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function.
Matching statistic: St001209
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001209: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Mp00066: Permutations —inverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001209: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,1,2] => [3,1,2] => 2
[2,1,3] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 3
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 2
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 3
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 2
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 3
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 4
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 4
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 3
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 3
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 4
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 3
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 4
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 3
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[1,5,3,2,4] => [4,2,3,5,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 4
[1,5,4,2,3] => [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 4
[2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[2,1,4,3,5] => [5,3,4,1,2] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[2,1,5,3,4] => [4,3,5,1,2] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 3
[2,3,1,4,5] => [5,4,1,3,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[2,3,4,1,5] => [5,1,4,3,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 4
[2,3,5,1,4] => [4,1,5,3,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 3
[2,4,1,3,5] => [5,3,1,4,2] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 2
[2,4,3,1,5] => [5,1,3,4,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 4
[2,4,5,1,3] => [3,1,5,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 3
[2,5,1,3,4] => [4,3,1,5,2] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 2
[2,5,3,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 4
[2,5,4,1,3] => [3,1,4,5,2] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 4
[3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 1
[3,1,4,2,5] => [5,2,4,1,3] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 3
[3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3
[3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 4
[3,2,4,1,5] => [5,1,4,2,3] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 3
[3,2,5,1,4] => [4,1,5,2,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 3
[3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[3,4,2,1,5] => [5,1,2,4,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 4
[3,4,5,1,2] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3
[3,5,1,2,4] => [4,2,1,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2
[3,5,2,1,4] => [4,1,2,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 4
[3,5,4,1,2] => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 4
[4,1,2,3,5] => [5,3,2,1,4] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 1
[4,1,3,2,5] => [5,2,3,1,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3
[4,1,5,2,3] => [3,2,5,1,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 3
[4,2,1,3,5] => [5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 4
[4,2,3,1,5] => [5,1,3,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3
[4,2,5,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 3
[4,3,1,2,5] => [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 4
[4,3,5,1,2] => [2,1,5,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[4,5,2,1,3] => [3,1,2,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 4
Description
The pmaj statistic of a parking function.
This is the parking function analogue of the bounce statistic, see [1, Section 1.3].
The definition given there is equivalent to the following: One again and again scans the parking function from right to left, keeping track of how often one has started over again. At step i, one marks the next position whose value is at most i with the number of restarts from the end of the parking function. The pmaj statistic is then the sum of the markings.
For example, consider the parking function [6,2,4,1,4,1,7,3].
In the first round, we mark positions 6,4,2 with 0's.
In the second round, we mark positions 8,5,3,1 with 1's.
In the third round, we mark position 7 with a 2.
In total, we obtain that
pmaj([6,2,4,1,4,1,7,3])=6.
This statistic is equidistributed with the area statistic [[St000188]] and the dinv statistic [[St000136]].
Matching statistic: St001433
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
St001433: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
St001433: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [-2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [-2,1,3] => 1
[2,3,1] => [2,3,1] => [2,3,1] => [-3,-2,1] => 2
[3,1,2] => [1,3,2] => [1,3,2] => [-3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => 3
[1,3,2,4] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [4,1,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => 2
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => 3
[2,4,1,3] => [2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => 2
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => 3
[3,4,1,2] => [1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => 2
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => 1
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => 3
[4,3,1,2] => [1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => [1,5,2,3,4] => ? = 4
[1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => [1,-4,2,3,5] => ? = 3
[1,2,4,5,3] => [4,1,2,5,3] => [4,1,2,5,3] => [1,-5,-4,2,3] => ? = 4
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,-5,2,3,4] => ? = 3
[1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[1,3,4,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => [4,-3,1,2,5] => ? = 3
[1,3,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => [5,-4,-3,1,2] => ? = 4
[1,3,5,2,4] => [3,1,2,5,4] => [3,1,2,5,4] => [5,-3,1,2,4] => ? = 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [4,1,2,3,5] => ? = 2
[1,4,3,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => [-4,3,1,2,5] => ? = 4
[1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => [5,-4,1,2,3] => ? = 3
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [5,1,2,3,4] => ? = 2
[1,5,3,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => [-5,3,1,2,4] => ? = 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => [-5,4,1,2,3] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => ? = 1
[2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => [-2,-4,1,3,5] => ? = 3
[2,1,5,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,1,3,4] => ? = 3
[2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => [-3,-2,1,4,5] => ? = 2
[2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => [-4,-3,-2,1,5] => ? = 3
[2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => [-5,-4,-3,-2,1] => ? = 4
[2,3,5,1,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-5,-3,-2,1,4] => ? = 3
[2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-4,-2,1,3,5] => ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => [2,3,-4,1,5] => ? = 4
[2,4,5,1,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-5,-4,-2,1,3] => ? = 3
[2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-5,-2,1,3,4] => ? = 2
[2,5,3,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [2,3,-5,1,4] => ? = 4
[2,5,4,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,4,-5,1,3] => ? = 4
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [-3,1,2,4,5] => ? = 1
[3,1,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 3
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [3,5,1,2,4] => ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [2,-3,1,4,5] => ? = 4
[3,2,4,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => [2,-4,-3,1,5] => ? = 3
[3,2,5,1,4] => [3,2,1,5,4] => [3,2,1,5,4] => [2,-5,-3,1,4] => ? = 3
[3,4,1,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => [-4,-3,1,2,5] => ? = 2
[3,4,2,1,5] => [3,4,2,1,5] => [3,4,2,1,5] => [2,4,-3,1,5] => ? = 4
[3,4,5,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => [-5,-4,-3,1,2] => ? = 3
[3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-5,-3,1,2,4] => ? = 2
[3,5,2,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [2,5,-3,1,4] => ? = 4
[3,5,4,1,2] => [1,5,3,4,2] => [1,5,3,4,2] => [3,4,-5,1,2] => ? = 4
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [-4,1,2,3,5] => ? = 1
[4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => [-3,-4,1,2,5] => ? = 3
[4,1,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => ? = 3
[4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => [2,-4,1,3,5] => ? = 4
[4,2,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => [3,-4,-2,1,5] => ? = 3
[4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => [2,-5,-4,1,3] => ? = 3
[4,3,1,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [3,-4,1,2,5] => ? = 4
[4,3,5,1,2] => [1,4,3,5,2] => [1,4,3,5,2] => [3,-5,-4,1,2] => ? = 3
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,4,5,3] => [-5,-4,1,2,3] => ? = 2
[4,5,2,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => [2,5,-4,1,3] => ? = 4
Description
The flag major index of a signed permutation.
The flag major index of a signed permutation σ is:
fmaj(σ)=neg(σ)+2⋅∑i∈DesB(σ)i,
where DesB(σ) is the B-descent set of σ; see [1, Eq.(10)].
This statistic is equidistributed with the B-inversions ([[St001428]]) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]).
Matching statistic: St001583
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Mp00064: Permutations —reverse⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [2,1,3] => 2
[3,1,2] => [1,3,2] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [2,3,4,1] => [1,4,3,2] => 3
[1,3,2,4] => [2,3,1,4] => [4,1,3,2] => 2
[1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 3
[1,4,2,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 2
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 3
[2,4,1,3] => [1,3,4,2] => [2,4,3,1] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,4,1,2] => [1,4,2,3] => [3,2,4,1] => 2
[4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[4,2,1,3] => [3,1,4,2] => [2,4,1,3] => 3
[4,3,1,2] => [1,4,3,2] => [2,3,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [2,3,4,5,1] => [1,5,4,3,2] => ? = 4
[1,2,4,3,5] => [2,3,4,1,5] => [5,1,4,3,2] => ? = 3
[1,2,4,5,3] => [2,3,5,1,4] => [4,1,5,3,2] => ? = 4
[1,2,5,3,4] => [2,3,1,5,4] => [4,5,1,3,2] => ? = 3
[1,3,2,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => ? = 2
[1,3,4,2,5] => [2,4,1,3,5] => [5,3,1,4,2] => ? = 3
[1,3,4,5,2] => [2,5,1,3,4] => [4,3,1,5,2] => ? = 4
[1,3,5,2,4] => [2,1,4,5,3] => [3,5,4,1,2] => ? = 3
[1,4,2,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => ? = 2
[1,4,3,2,5] => [3,4,2,1,5] => [5,1,2,4,3] => ? = 4
[1,4,5,2,3] => [2,1,5,3,4] => [4,3,5,1,2] => ? = 3
[1,5,2,3,4] => [2,1,3,5,4] => [4,5,3,1,2] => ? = 2
[1,5,3,2,4] => [3,4,1,5,2] => [2,5,1,4,3] => ? = 4
[1,5,4,2,3] => [3,1,5,4,2] => [2,4,5,1,3] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => ? = 1
[2,1,4,3,5] => [3,2,4,1,5] => [5,1,4,2,3] => ? = 3
[2,1,5,3,4] => [3,2,1,5,4] => [4,5,1,2,3] => ? = 3
[2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => ? = 2
[2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => ? = 3
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => ? = 4
[2,3,5,1,4] => [1,3,4,5,2] => [2,5,4,3,1] => ? = 3
[2,4,1,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => ? = 4
[2,4,5,1,3] => [1,3,5,2,4] => [4,2,5,3,1] => ? = 3
[2,5,1,3,4] => [1,3,2,5,4] => [4,5,2,3,1] => ? = 2
[2,5,3,1,4] => [4,2,1,5,3] => [3,5,1,2,4] => ? = 4
[2,5,4,1,3] => [1,4,5,3,2] => [2,3,5,4,1] => ? = 4
[3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
[3,1,4,2,5] => [3,4,1,2,5] => [5,2,1,4,3] => ? = 3
[3,1,5,2,4] => [3,1,4,5,2] => [2,5,4,1,3] => ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ? = 4
[3,2,4,1,5] => [4,2,1,3,5] => [5,3,1,2,4] => ? = 3
[3,2,5,1,4] => [1,4,3,5,2] => [2,5,3,4,1] => ? = 3
[3,4,1,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 2
[3,4,2,1,5] => [4,3,1,2,5] => [5,2,1,3,4] => ? = 4
[3,4,5,1,2] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 3
[3,5,1,2,4] => [1,2,4,5,3] => [3,5,4,2,1] => ? = 2
[3,5,2,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => ? = 4
[3,5,4,1,2] => [1,5,3,4,2] => [2,4,3,5,1] => ? = 4
[4,1,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
[4,1,3,2,5] => [2,4,3,1,5] => [5,1,3,4,2] => ? = 3
[4,1,5,2,3] => [3,1,5,2,4] => [4,2,5,1,3] => ? = 3
[4,2,1,3,5] => [3,1,4,2,5] => [5,2,4,1,3] => ? = 4
[4,2,3,1,5] => [4,1,3,2,5] => [5,2,3,1,4] => ? = 3
[4,2,5,1,3] => [1,4,5,2,3] => [3,2,5,4,1] => ? = 3
[4,3,1,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => ? = 4
[4,3,5,1,2] => [1,5,3,2,4] => [4,2,3,5,1] => ? = 3
[4,5,1,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 2
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 43%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> ? = 0
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[2,1] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 3
[3,4,1,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[4,1,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 3
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4
[1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 3
[1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 4
[1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3
[1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 3
[1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 4
[1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 3
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 4
[1,4,5,2,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 3
[1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[1,5,3,2,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 4
[1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[2,1,5,3,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 3
[2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 3
[2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4
[2,3,5,1,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 3
[2,4,1,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 4
[2,4,5,1,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 3
[2,5,1,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2
[2,5,3,1,4] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> ? = 4
[2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ? = 4
[3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1
[3,1,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[3,1,5,2,4] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 4
[3,2,4,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[3,2,5,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? = 3
[3,4,1,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[3,4,2,1,5] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 4
[3,4,5,1,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3
[3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[3,5,2,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ? = 4
[3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 4
[4,1,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[4,1,3,2,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 3
[4,1,5,2,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3
[4,2,1,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 4
[4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 3
[4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 3
[4,3,1,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 4
[5,1,2,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
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St001645The pebbling number of a connected graph.
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