Your data matches 75 different statistics following compositions of up to 3 maps.
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Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[8,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001000
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
St001000: Dyck paths ⟶ ℤResult quality: 45% values known / values provided: 47%distinct values known / distinct values provided: 45%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[8,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 7
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 4
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[5,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 7
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 9
[3,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 4
[7,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[6,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[5,5,2]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[5,5,1,1]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 7
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 7
[4,4,4]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 9
[4,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 4
[3,3,3,3]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
[8,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[8,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[7,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[6,6,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[6,5,2]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[6,5,1,1]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 7
[6,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 7
[5,5,3]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[5,5,1,1,1]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 7
[5,4,4]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[5,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[5,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 9
[5,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 4
[4,4,4,1]
=> [4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 8
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 8
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00064: Permutations reversePermutations
St000672: Permutations ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [3,1,2] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 2
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 3
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 4
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [6,5,2,1,4,3] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [6,5,4,1,2,3] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [6,5,1,2,3,4] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [6,1,2,3,4,5] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 5
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [7,6,5,2,1,4,3] => ? = 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [7,6,5,4,1,2,3] => 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ? = 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [7,6,5,1,2,3,4] => 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [4,3,1,7,6,2,5] => ? = 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [7,2,1,4,3,6,5] => ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [7,6,1,2,3,4,5] => 4
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [7,1,2,3,4,5,6] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 6
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [8,7,6,5,2,1,4,3] => 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [8,7,6,5,4,1,2,3] => 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [8,7,3,2,1,6,5,4] => ? = 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [8,7,6,5,1,2,3,4] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [8,4,3,1,7,6,2,5] => 3
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [8,7,2,1,4,3,6,5] => 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [8,7,6,1,2,3,4,5] => 4
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [8,7,1,2,3,4,5,6] => 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 3
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [8,1,2,3,4,5,6,7] => 6
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,1,2] => 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [9,8,7,6,5,2,1,4,3] => ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [9,8,7,6,5,4,1,2,3] => 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [9,8,7,3,2,1,6,5,4] => ? = 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [9,8,7,6,5,1,2,3,4] => 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [9,4,3,2,1,8,7,6,5] => ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [9,8,4,3,1,7,6,2,5] => ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [9,8,7,2,1,4,3,6,5] => ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [9,8,7,6,1,2,3,4,5] => 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [5,4,3,1,9,8,7,2,6] => ? = 4
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [9,8,7,1,2,3,4,5,6] => 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? = 2
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [6,5,1,9,8,2,3,4,7] => ? = 5
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [9,2,1,4,3,6,5,8,7] => ? = 3
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [9,8,1,2,3,4,5,6,7] => 6
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,1,2] => 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [10,9,8,7,6,5,2,1,4,3] => ? = 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,1,2,3] => 2
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [10,9,8,7,3,2,1,6,5,4] => ? = 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [10,9,8,7,6,5,1,2,3,4] => 3
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [10,9,4,3,2,1,8,7,6,5] => ? = 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [10,9,8,4,3,1,7,6,2,5] => ? = 3
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [10,9,8,7,2,1,4,3,6,5] => ? = 2
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [10,9,8,7,6,1,2,3,4,5] => 4
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => ? = 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [10,5,4,3,1,9,8,7,2,6] => ? = 4
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => [10,9,8,7,1,2,3,4,5,6] => 5
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [6,5,2,1,10,9,4,3,8,7] => ? = 3
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [6,5,4,1,10,9,8,2,3,7] => ? = 5
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [10,3,2,1,6,5,4,9,8,7] => ? = 2
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => [10,6,5,1,9,8,2,3,4,7] => ? = 5
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10] => [10,9,2,1,4,3,6,5,8,7] => ? = 3
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => [10,9,8,1,2,3,4,5,6,7] => 6
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [8,5,4,3,2,9,10,1,6,7] => [7,6,1,10,9,2,3,4,5,8] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,10,9] => 4
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? => ? = 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? => ? = 2
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? => ? = 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? => ? = 3
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? => ? = 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? => ? = 3
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? => ? = 2
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? => ? = 4
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 4
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 5
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => [11,6,5,2,1,10,9,4,3,8,7] => ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => [11,6,5,4,1,10,9,8,2,3,7] => ? = 5
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => [11,10,3,2,1,6,5,4,9,8,7] => ? = 2
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => [11,10,6,5,1,9,8,2,3,4,7] => ? = 5
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => [11,10,9,2,1,4,3,6,5,8,7] => ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? => ? = 6
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 7
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? => ? = 9
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? => ? = 4
[11,1]
=> [[1,3,4,5,6,7,8,9,10,11,12],[2]]
=> ? => ? => ? = 1
Description
The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is $$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$ for some $(r,a,b)$. This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
St000662: Permutations ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,1,5,3,6,7,4] => ? = 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,3,5,6,4,2,7] => 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 4
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 6
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,4,2,5,6,7,8] => ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,4,5,6,3,7,8] => ? = 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,1,5,3,6,7,4,8] => ? = 3
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,3,5,6,4,2,7,8] => ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 4
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => 3
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 6
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,4,2,5,6,7,8,9] => ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,4,5,6,3,7,8,9] => ? = 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,5,6,7,8,4,9] => ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,1,5,3,6,7,4,8,9] => ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,3,5,6,4,2,7,8,9] => ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,1,3,6,4,7,8,9,5] => ? = 4
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => ? = 2
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [4,3,7,2,1,5,8,9,6] => ? = 5
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,3,5,7,8,6,4,2,9] => ? = 3
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 6
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,3,4,2,5,6,7,8,9,10] => ? = 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 2
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,4,5,6,3,7,8,9,10] => ? = 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 3
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,5,6,7,8,4,9,10] => ? = 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [2,1,5,3,6,7,4,8,9,10] => ? = 3
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,3,5,6,4,2,7,8,9,10] => ? = 2
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 4
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,6,7,8,9,10,5] => ? = 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [2,1,3,6,4,7,8,9,5,10] => ? = 4
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 5
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [1,3,4,2,7,8,5,9,10,6] => ? = 3
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [3,2,1,7,4,5,8,9,10,6] => ? = 5
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [1,2,4,5,7,8,9,6,3,10] => ? = 2
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => [4,3,7,2,1,5,8,9,6,10] => ? = 5
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10] => [1,3,5,7,8,6,4,2,9,10] => ? = 3
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 6
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [8,5,4,3,2,9,10,1,6,7] => [5,4,8,3,2,1,6,9,10,7] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [1,3,5,7,9,10,8,6,4,2] => ? = 4
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? => ? = 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? => ? = 2
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? => ? = 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? => ? = 3
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? => ? = 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? => ? = 3
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? => ? = 2
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? => ? = 4
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 4
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 5
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => [1,3,4,2,7,8,5,9,10,6,11] => ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => [3,2,1,7,4,5,8,9,10,6,11] => ? = 5
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => [1,2,4,5,7,8,9,6,3,10,11] => ? = 2
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => [4,3,7,2,1,5,8,9,6,10,11] => ? = 5
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => [1,3,5,7,8,6,4,2,9,10,11] => ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? => ? = 6
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 7
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? => ? = 9
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? => ? = 4
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000366
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
St000366: Permutations ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1 = 2 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3 = 4 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 0 = 1 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 4 = 5 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0 = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => 0 = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1 = 2 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => 0 = 1 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,1,5,3,6,7,4] => ? = 3 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,3,5,6,4,2,7] => 1 = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5 = 6 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 0 = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,4,2,5,6,7,8] => ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 1 = 2 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,4,5,6,3,7,8] => ? = 1 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 2 = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,1,5,3,6,7,4,8] => ? = 3 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,3,5,6,4,2,7,8] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 3 = 4 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 4 = 5 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => 2 = 3 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 5 = 6 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 0 = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,4,2,5,6,7,8,9] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 1 = 2 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,4,5,6,3,7,8,9] => ? = 1 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 2 = 3 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,5,6,7,8,4,9] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,1,5,3,6,7,4,8,9] => ? = 3 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,3,5,6,4,2,7,8,9] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 3 = 4 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,1,3,6,4,7,8,9,5] => ? = 4 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 4 = 5 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => ? = 2 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [4,3,7,2,1,5,8,9,6] => ? = 5 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,3,5,7,8,6,4,2,9] => ? = 3 - 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 5 = 6 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,3,4,2,5,6,7,8,9,10] => ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 1 = 2 - 1
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,4,5,6,3,7,8,9,10] => ? = 1 - 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 2 = 3 - 1
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,5,6,7,8,4,9,10] => ? = 1 - 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [2,1,5,3,6,7,4,8,9,10] => ? = 3 - 1
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,3,5,6,4,2,7,8,9,10] => ? = 2 - 1
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 3 = 4 - 1
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,6,7,8,9,10,5] => ? = 1 - 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [2,1,3,6,4,7,8,9,5,10] => ? = 4 - 1
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 4 = 5 - 1
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [1,3,4,2,7,8,5,9,10,6] => ? = 3 - 1
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [3,2,1,7,4,5,8,9,10,6] => ? = 5 - 1
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [1,2,4,5,7,8,9,6,3,10] => ? = 2 - 1
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => [4,3,7,2,1,5,8,9,6,10] => ? = 5 - 1
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10] => [1,3,5,7,8,6,4,2,9,10] => ? = 3 - 1
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 5 = 6 - 1
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [8,5,4,3,2,9,10,1,6,7] => [5,4,8,3,2,1,6,9,10,7] => ? = 7 - 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [1,3,5,7,9,10,8,6,4,2] => ? = 4 - 1
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 1 - 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? => ? = 1 - 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? => ? = 2 - 1
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? => ? = 1 - 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? => ? = 3 - 1
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? => ? = 1 - 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? => ? = 3 - 1
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? => ? = 2 - 1
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? => ? = 4 - 1
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 1 - 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 4 - 1
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 5 - 1
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 5 - 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => [1,3,4,2,7,8,5,9,10,6,11] => ? = 3 - 1
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => [3,2,1,7,4,5,8,9,10,6,11] => ? = 5 - 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => [1,2,4,5,7,8,9,6,3,10,11] => ? = 2 - 1
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => [4,3,7,2,1,5,8,9,6,10,11] => ? = 5 - 1
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => [1,3,5,7,8,6,4,2,9,10,11] => ? = 3 - 1
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? => ? = 6 - 1
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 7 - 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? => ? = 9 - 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? => ? = 4 - 1
Description
The number of double descents of a permutation. A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
St000371: Permutations ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1 = 2 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3 = 4 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 0 = 1 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 4 = 5 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0 = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => 0 = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1 = 2 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => 0 = 1 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,1,5,3,6,7,4] => ? = 3 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,3,5,6,4,2,7] => 1 = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5 = 6 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 0 = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,4,2,5,6,7,8] => ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 1 = 2 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,4,5,6,3,7,8] => ? = 1 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 2 = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,1,5,3,6,7,4,8] => ? = 3 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,3,5,6,4,2,7,8] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 3 = 4 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 4 = 5 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => 2 = 3 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 5 = 6 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 0 = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,4,2,5,6,7,8,9] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 1 = 2 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,4,5,6,3,7,8,9] => ? = 1 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 2 = 3 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,5,6,7,8,4,9] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,1,5,3,6,7,4,8,9] => ? = 3 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,3,5,6,4,2,7,8,9] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 3 = 4 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,1,3,6,4,7,8,9,5] => ? = 4 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 4 = 5 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => ? = 2 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [4,3,7,2,1,5,8,9,6] => ? = 5 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,3,5,7,8,6,4,2,9] => ? = 3 - 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 5 = 6 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,3,4,2,5,6,7,8,9,10] => ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 1 = 2 - 1
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,4,5,6,3,7,8,9,10] => ? = 1 - 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 2 = 3 - 1
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,5,6,7,8,4,9,10] => ? = 1 - 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [2,1,5,3,6,7,4,8,9,10] => ? = 3 - 1
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,3,5,6,4,2,7,8,9,10] => ? = 2 - 1
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 3 = 4 - 1
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,6,7,8,9,10,5] => ? = 1 - 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [2,1,3,6,4,7,8,9,5,10] => ? = 4 - 1
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 4 = 5 - 1
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [1,3,4,2,7,8,5,9,10,6] => ? = 3 - 1
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [3,2,1,7,4,5,8,9,10,6] => ? = 5 - 1
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [1,2,4,5,7,8,9,6,3,10] => ? = 2 - 1
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => [4,3,7,2,1,5,8,9,6,10] => ? = 5 - 1
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10] => [1,3,5,7,8,6,4,2,9,10] => ? = 3 - 1
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 5 = 6 - 1
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [8,5,4,3,2,9,10,1,6,7] => [5,4,8,3,2,1,6,9,10,7] => ? = 7 - 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [1,3,5,7,9,10,8,6,4,2] => ? = 4 - 1
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 1 - 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? => ? = 1 - 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? => ? = 2 - 1
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? => ? = 1 - 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? => ? = 3 - 1
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? => ? = 1 - 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? => ? = 3 - 1
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? => ? = 2 - 1
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? => ? = 4 - 1
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 1 - 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 4 - 1
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 5 - 1
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 5 - 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => [1,3,4,2,7,8,5,9,10,6,11] => ? = 3 - 1
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => [3,2,1,7,4,5,8,9,10,6,11] => ? = 5 - 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => [1,2,4,5,7,8,9,6,3,10,11] => ? = 2 - 1
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => [4,3,7,2,1,5,8,9,6,10,11] => ? = 5 - 1
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => [1,3,5,7,8,6,4,2,9,10,11] => ? = 3 - 1
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? => ? = 6 - 1
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 7 - 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? => ? = 9 - 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? => ? = 4 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].
Matching statistic: St000074
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
Mp00082: Standard tableaux to Gelfand-Tsetlin patternGelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
[3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 5
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> [[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7]]
=> [[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7]]
=> [[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> [[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> [[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> [[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 6
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[2,1,1,1,1,1,1,0],[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> [[2,2,1,1,1,1,0,0],[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[3,1,1,1,1,1,0,0],[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> [[2,2,2,1,1,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[4,1,1,1,1,0,0,0],[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8]]
=> [[3,2,2,1,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> [[3,3,1,1,0,0,0,0],[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[5,1,1,1,0,0,0,0],[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[6,1,1,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 3
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7],[8]]
=> [[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 6
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[2,1,1,1,1,1,1,1,0],[2,1,1,1,1,1,1,0],[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> [[2,2,1,1,1,1,1,0,0],[2,2,1,1,1,1,0,0],[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[3,1,1,1,1,1,1,0,0],[3,1,1,1,1,1,0,0],[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> [[2,2,2,1,1,1,0,0,0],[2,2,2,1,1,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[4,1,1,1,1,1,0,0,0],[4,1,1,1,1,0,0,0],[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> [[2,2,2,2,1,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8],[9]]
=> [[3,2,2,1,1,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> [[3,3,1,1,1,0,0,0,0],[3,3,1,1,0,0,0,0],[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[5,1,1,1,1,0,0,0,0],[5,1,1,1,0,0,0,0],[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [[1,2,6],[3,7],[4,8],[5,9]]
=> [[3,2,2,2,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 4
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[6,1,1,1,0,0,0,0,0],[6,1,1,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [[3,3,3,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [[1,2,3,4,7],[5,8],[6,9]]
=> [[5,2,2,0,0,0,0,0,0],[5,2,1,0,0,0,0,0],[5,1,1,0,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> [[4,4,1,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 3
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[7,1,1,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 6
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[2,1,1,1,1,1,1,1,1,0],[2,1,1,1,1,1,1,1,0],[2,1,1,1,1,1,1,0],[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> [[2,2,1,1,1,1,1,1,0,0],[2,2,1,1,1,1,1,0,0],[2,2,1,1,1,1,0,0],[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [[3,1,1,1,1,1,1,1,0,0],[3,1,1,1,1,1,1,0,0],[3,1,1,1,1,1,0,0],[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10]]
=> [[2,2,2,1,1,1,1,0,0,0],[2,2,2,1,1,1,0,0,0],[2,2,2,1,1,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [[4,1,1,1,1,1,1,0,0,0],[4,1,1,1,1,1,0,0,0],[4,1,1,1,1,0,0,0],[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9],[10]]
=> [[2,2,2,2,1,1,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8],[9],[10]]
=> [[3,2,2,1,1,1,0,0,0,0],[3,2,2,1,1,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 3
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> [[3,3,1,1,1,1,0,0,0,0],[3,3,1,1,1,0,0,0,0],[3,3,1,1,0,0,0,0],[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 2
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> [[5,1,1,1,1,1,0,0,0,0],[5,1,1,1,1,0,0,0,0],[5,1,1,1,0,0,0,0],[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [[2,2,2,2,2,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [[1,2,6],[3,7],[4,8],[5,9],[10]]
=> [[3,2,2,2,1,0,0,0,0,0],[3,2,2,2,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 4
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [[6,1,1,1,1,0,0,0,0,0],[6,1,1,1,0,0,0,0,0],[6,1,1,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 5
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10]]
=> [[3,3,2,2,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 3
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10]]
=> [[4,2,2,2,0,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[4,2,1,1,0,0,0,0],[4,1,1,1,0,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> [[3,3,3,1,0,0,0,0,0,0],[3,3,3,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [[1,2,3,4,7],[5,8],[6,9],[10]]
=> [[5,2,2,1,0,0,0,0,0,0],[5,2,2,0,0,0,0,0,0],[5,2,1,0,0,0,0,0],[5,1,1,0,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> [[4,4,1,1,0,0,0,0,0,0],[4,4,1,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 3
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [[7,1,1,1,0,0,0,0,0,0],[7,1,1,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 6
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [[1,2,3,4,5,8],[6,9],[7,10]]
=> [[6,2,2,0,0,0,0,0,0,0],[6,2,1,0,0,0,0,0,0],[6,1,1,0,0,0,0,0],[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 4
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ?
=> ?
=> ? = 2
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ?
=> ? = 3
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ?
=> ?
=> ? = 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ?
=> ?
=> ? = 3
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 2
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ?
=> ?
=> ? = 4
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10],[11]]
=> ?
=> ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ?
=> ?
=> ? = 4
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ?
=> ?
=> ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
=> ?
=> ? = 5
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> [[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,2,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> [[4,2,2,2,1,0,0,0,0,0,0],[4,2,2,2,0,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[4,2,1,1,0,0,0,0],[4,1,1,1,0,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> [[3,3,3,1,1,0,0,0,0,0,0],[3,3,3,1,0,0,0,0,0,0],[3,3,3,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [[1,2,3,4,7],[5,8],[6,9],[10],[11]]
=> [[5,2,2,1,1,0,0,0,0,0,0],[5,2,2,1,0,0,0,0,0,0],[5,2,2,0,0,0,0,0,0],[5,2,1,0,0,0,0,0],[5,1,1,0,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> [[4,4,1,1,1,0,0,0,0,0,0],[4,4,1,1,0,0,0,0,0,0],[4,4,1,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ?
=> ?
=> ? = 6
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 5
Description
The number of special entries. An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000176
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
Mp00082: Standard tableaux to Gelfand-Tsetlin patternGelfand-Tsetlin patterns
St000176: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [[2,0],[1]]
=> 3 = 1 + 2
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 3 = 1 + 2
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 4 = 2 + 2
[3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> 3 = 1 + 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 4 = 2 + 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 5 = 3 + 2
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 4 = 2 + 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 5 = 3 + 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 6 = 4 + 2
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> [[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4 = 2 + 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 5 = 3 + 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 4 = 2 + 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 6 = 4 + 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 7 = 5 + 2
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4 = 2 + 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 5 = 3 + 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [[3,3,1,0,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [[5,1,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 6 = 4 + 2
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 5 + 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 8 = 6 + 2
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [[2,1,1,1,1,1,1,0],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [[2,2,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [[3,1,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4 = 2 + 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [[4,1,1,1,1,0,0,0],[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 5 = 3 + 2
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [[5,1,1,1,0,0,0,0],[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 6 = 4 + 2
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[6,1,1,0,0,0,0,0],[5,1,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 7 = 5 + 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 5 = 3 + 2
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 8 = 6 + 2
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [[2,1,1,1,1,1,1,1,0],[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[1,8],[2,9],[3],[4],[5],[6],[7]]
=> [[2,2,1,1,1,1,1,0,0],[2,1,1,1,1,1,1,0],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [[3,1,1,1,1,1,1,0,0],[2,1,1,1,1,1,1,0],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4 = 2 + 2
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> [[2,2,2,1,1,1,0,0,0],[2,2,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [[4,1,1,1,1,1,0,0,0],[3,1,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 5 = 3 + 2
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> [[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [[3,2,2,1,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [[3,3,1,1,1,0,0,0,0],[3,2,1,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [[5,1,1,1,1,0,0,0,0],[4,1,1,1,1,0,0,0],[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 6 = 4 + 2
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [[3,2,2,2,0,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 + 2
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [[6,1,1,1,0,0,0,0,0],[5,1,1,1,0,0,0,0],[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 7 = 5 + 2
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [[3,3,3,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> [[5,2,2,0,0,0,0,0,0],[4,2,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 5 + 2
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> [[4,4,1,0,0,0,0,0,0],[4,3,1,0,0,0,0,0],[3,3,1,0,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [[7,1,1,0,0,0,0,0,0],[6,1,1,0,0,0,0,0],[5,1,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 8 = 6 + 2
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [[2,1,1,1,1,1,1,1,1,0],[1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 3 = 1 + 2
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]
=> [[2,2,1,1,1,1,1,1,0,0],[2,1,1,1,1,1,1,1,0],[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> [[3,1,1,1,1,1,1,1,0,0],[2,1,1,1,1,1,1,1,0],[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4 = 2 + 2
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [[1,8],[2,9],[3,10],[4],[5],[6],[7]]
=> [[2,2,2,1,1,1,1,0,0,0],[2,2,1,1,1,1,1,0,0],[2,1,1,1,1,1,1,0],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [[4,1,1,1,1,1,1,0,0,0],[3,1,1,1,1,1,1,0,0],[2,1,1,1,1,1,1,0],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 5 = 3 + 2
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> [[2,2,2,2,1,1,0,0,0,0],[2,2,2,1,1,1,0,0,0],[2,2,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [[1,7,10],[2,8],[3,9],[4],[5],[6]]
=> [[3,2,2,1,1,1,0,0,0,0],[2,2,2,1,1,1,0,0,0],[2,2,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [[1,7,9],[2,8,10],[3],[4],[5],[6]]
=> [[3,3,1,1,1,1,0,0,0,0],[3,2,1,1,1,1,0,0,0],[2,2,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [[5,1,1,1,1,1,0,0,0,0],[4,1,1,1,1,1,0,0,0],[3,1,1,1,1,1,0,0],[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 6 = 4 + 2
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [[2,2,2,2,2,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 + 2
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [[1,6,10],[2,7],[3,8],[4,9],[5]]
=> [[3,2,2,2,1,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 4 + 2
[5,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [[6,1,1,1,1,0,0,0,0,0],[5,1,1,1,1,0,0,0,0],[4,1,1,1,1,0,0,0],[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 7 = 5 + 2
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [[1,5,9],[2,6,10],[3,7],[4,8]]
=> [[3,3,2,2,0,0,0,0,0,0],[3,2,2,2,0,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [[1,5,9,10],[2,6],[3,7],[4,8]]
=> [[4,2,2,2,0,0,0,0,0,0],[3,2,2,2,0,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 5 + 2
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [[1,5,8],[2,6,9],[3,7,10],[4]]
=> [[3,3,3,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [[1,5,8,9,10],[2,6],[3,7],[4]]
=> [[5,2,2,1,0,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 5 + 2
[4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [[1,5,7,9],[2,6,8,10],[3],[4]]
=> [[4,4,1,1,0,0,0,0,0,0],[4,3,1,1,0,0,0,0,0],[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[4,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [[7,1,1,1,0,0,0,0,0,0],[6,1,1,1,0,0,0,0,0],[5,1,1,1,0,0,0,0],[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 8 = 6 + 2
[3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [[1,4,7,8,9,10],[2,5],[3,6]]
=> [[6,2,2,0,0,0,0,0,0,0],[5,2,2,0,0,0,0,0,0],[4,2,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 7 + 2
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 6 = 4 + 2
[10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ?
=> ? = 1 + 2
[9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> [[1,10],[2,11],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 1 + 2
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 2 + 2
[8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> [[1,9],[2,10],[3,11],[4],[5],[6],[7],[8]]
=> ?
=> ? = 1 + 2
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ?
=> ? = 3 + 2
[7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ?
=> ?
=> ? = 1 + 2
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> [[1,8,11],[2,9],[3,10],[4],[5],[6],[7]]
=> ?
=> ? = 3 + 2
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> [[1,8,10],[2,9,11],[3],[4],[5],[6],[7]]
=> ?
=> ? = 2 + 2
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ?
=> ?
=> ? = 4 + 2
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6]]
=> ?
=> ? = 1 + 2
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ?
=> ? = 4 + 2
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ?
=> ? = 5 + 2
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> [[1,6,11],[2,7],[3,8],[4,9],[5,10]]
=> ?
=> ? = 5 + 2
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [[1,6,10],[2,7,11],[3,8],[4,9],[5]]
=> [[3,3,2,2,1,0,0,0,0,0,0],[3,2,2,2,1,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [[1,6,10,11],[2,7],[3,8],[4,9],[5]]
=> [[4,2,2,2,1,0,0,0,0,0,0],[3,2,2,2,1,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 5 + 2
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [[1,6,9],[2,7,10],[3,8,11],[4],[5]]
=> [[3,3,3,1,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,2,2,1,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 2 + 2
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [[1,6,9,10,11],[2,7],[3,8],[4],[5]]
=> [[5,2,2,1,1,0,0,0,0,0,0],[4,2,2,1,1,0,0,0,0,0],[3,2,2,1,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 5 + 2
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [[1,6,8,10],[2,7,9,11],[3],[4],[5]]
=> [[4,4,1,1,1,0,0,0,0,0,0],[4,3,1,1,1,0,0,0,0,0],[3,3,1,1,1,0,0,0,0],[3,2,1,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 3 + 2
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ?
=> ? = 6 + 2
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 7 = 5 + 2
Description
The total number of tiles in the Gelfand-Tsetlin pattern. The tiling of a Gelfand-Tsetlin pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belongs to exactly one tile.
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
St000141: Permutations ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 55%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => ? = 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => ? = 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,1,5,3,6,7,4] => ? = 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,3,5,6,4,2,7] => ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 4
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 6
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,4,2,5,6,7,8] => ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,4,5,6,3,7,8] => ? = 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,1,5,3,6,7,4,8] => ? = 3
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,3,5,6,4,2,7,8] => ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 4
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => 3
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 6
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,4,2,5,6,7,8,9] => ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,4,5,6,3,7,8,9] => ? = 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,5,6,7,8,4,9] => ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,1,5,3,6,7,4,8,9] => ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,3,5,6,4,2,7,8,9] => ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,1,3,6,4,7,8,9,5] => ? = 4
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => ? = 2
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [4,3,7,2,1,5,8,9,6] => ? = 5
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,3,5,7,8,6,4,2,9] => ? = 3
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 6
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,3,4,2,5,6,7,8,9,10] => ? = 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 2
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,4,5,6,3,7,8,9,10] => ? = 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 3
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,5,6,7,8,4,9,10] => ? = 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [2,1,5,3,6,7,4,8,9,10] => ? = 3
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,3,5,6,4,2,7,8,9,10] => ? = 2
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 4
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,6,7,8,9,10,5] => ? = 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [2,1,3,6,4,7,8,9,5,10] => ? = 4
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 5
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [1,3,4,2,7,8,5,9,10,6] => ? = 3
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [3,2,1,7,4,5,8,9,10,6] => ? = 5
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [1,2,4,5,7,8,9,6,3,10] => ? = 2
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => [4,3,7,2,1,5,8,9,6,10] => ? = 5
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10] => [1,3,5,7,8,6,4,2,9,10] => ? = 3
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 6
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [8,5,4,3,2,9,10,1,6,7] => [5,4,8,3,2,1,6,9,10,7] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [1,3,5,7,9,10,8,6,4,2] => ? = 4
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? => ? = 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? => ? = 2
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? => ? = 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? => ? = 3
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? => ? = 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? => ? = 3
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? => ? = 2
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? => ? = 4
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 4
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 5
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => [1,3,4,2,7,8,5,9,10,6,11] => ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => [3,2,1,7,4,5,8,9,10,6,11] => ? = 5
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => [1,2,4,5,7,8,9,6,3,10,11] => ? = 2
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => [4,3,7,2,1,5,8,9,6,10,11] => ? = 5
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => [1,3,5,7,8,6,4,2,9,10,11] => ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? => ? = 6
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St001491
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00200: Binary words twistBinary words
St001491: Binary words ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 18%
Values
[1,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[2,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[1,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[3,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[2,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[2,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[4,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[3,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[3,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 4 - 1
[5,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[4,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[4,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[3,3]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[2,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 5 - 1
[6,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[5,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[5,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[4,3]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[3,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 3 - 1
[3,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 5 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? = 6 - 1
[7,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[6,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[6,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[5,3]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[4,4]
=> [4]
=> 10000 => 00000 => ? = 1 - 1
[4,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 3 - 1
[4,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 4 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 5 - 1
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01100 => ? = 3 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? = 6 - 1
[8,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[7,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[7,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[6,3]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[6,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[5,4]
=> [4]
=> 10000 => 00000 => ? = 1 - 1
[5,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 3 - 1
[5,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[5,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 4 - 1
[4,4,1]
=> [4,1]
=> 100010 => 000010 => ? = 4 - 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 5 - 1
[3,3,3]
=> [3,3]
=> 11000 => 01000 => ? = 2 - 1
[3,3,1,1,1]
=> [3,1,1,1]
=> 1001110 => 0001110 => ? = 5 - 1
[3,2,2,2]
=> [2,2,2]
=> 11100 => 01100 => ? = 3 - 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? = 6 - 1
[9,1]
=> [1]
=> 10 => 00 => ? = 1 - 1
[8,2]
=> [2]
=> 100 => 000 => ? = 1 - 1
[8,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[7,3]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[7,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[6,4]
=> [4]
=> 10000 => 00000 => ? = 1 - 1
[6,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 3 - 1
[6,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[6,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 4 - 1
[5,5]
=> [5]
=> 100000 => 000000 => ? = 1 - 1
[5,4,1]
=> [4,1]
=> 100010 => 000010 => ? = 4 - 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 5 - 1
[4,4,2]
=> [4,2]
=> 100100 => 000100 => ? = 3 - 1
[9,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[8,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[7,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[10,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[9,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[8,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[11,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[10,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[9,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[12,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[11,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[10,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[13,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[12,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[11,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[14,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[13,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[12,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[15,1,1]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[14,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[13,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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St000071The number of maximal chains in a poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000068The number of minimal elements in a poset. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000909The number of maximal chains of maximal size in a poset. St000211The rank of the set partition. St000245The number of ascents of a permutation. St000363The number of minimal vertex covers of a graph. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000527The width of the poset. St001304The number of maximally independent sets of vertices of a graph. St001458The rank of the adjacency matrix of a graph. St000910The number of maximal chains of minimal length in a poset. St000632The jump number of the poset. St000740The last entry of a permutation. St000354The number of recoils of a permutation. St000308The height of the tree associated to a permutation. St001427The number of descents of a signed permutation. St000006The dinv of a Dyck path. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001330The hat guessing number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000021The number of descents of a permutation. St000619The number of cyclic descents of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000325The width of the tree associated to a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000039The number of crossings of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000822The Hadwiger number of the graph. St000990The first ascent of a permutation.