Processing math: 100%

Your data matches 11 different statistics following compositions of up to 3 maps.
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Mp00071: Permutations descent compositionInteger compositions
Mp00173: Integer compositions rotate front to backInteger compositions
Mp00231: Integer compositions bounce pathDyck 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 compositionInteger compositions
Mp00173: Integer compositions rotate front to backInteger compositions
Mp00231: Integer compositions bounce pathDyck 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.
Mp00069: Permutations complementPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00170: Permutations to signed permutationSigned 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 1i<jn such that 0<π(i)<π(j).
Matching statistic: St001931
Mp00305: Permutations parking functionParking functions
Mp00319: Parking functions to compositionInteger 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), 1i<ncici+1i.
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00064: Permutations reversePermutations
Mp00305: Permutations parking functionParking 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.
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00064: Permutations reversePermutations
Mp00305: Permutations parking functionParking 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 reversePermutations
Mp00066: Permutations inversePermutations
Mp00305: Permutations parking functionParking 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]].
Mp00067: Permutations Foata bijectionPermutations
Mp00170: Permutations to signed permutationSigned permutations
Mp00194: Signed permutations Foata-Han inverseSigned 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)+2iDesB(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]).
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00064: Permutations reversePermutations
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.
Mp00067: Permutations Foata bijectionPermutations
Mp00065: Permutations permutation posetPosets
Mp00195: Posets order idealsLattices
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.