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Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [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 removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 47%●distinct values known / distinct values provided: 45%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck 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.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
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].
Matching statistic: St000662
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
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 tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
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 tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
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 tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 55%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-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 tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000176: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 55%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-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.
Matching statistic: St000141
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 55%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
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 removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 18%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary 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.
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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.
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