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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 0
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 0
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 0
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 0
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 0
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 5
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 5
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000352
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => 3 = 0 + 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => 3 = 0 + 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => 3 = 0 + 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => 3 = 0 + 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => 3 = 0 + 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => 3 = 0 + 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => 3 = 0 + 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => 3 = 0 + 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => 3 = 0 + 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => 4 = 1 + 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => 3 = 0 + 3
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => 3 = 0 + 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => 3 = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> [15,13,14,10,11,12,5,6,7,8,1,2,3,4,9] => ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [15,13,14,9,10,11,6,7,8,12,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [15,13,14,9,10,11,5,6,7,12,1,2,3,4,8] => ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [15,13,14,8,9,10,5,6,7,11,1,2,3,4,12] => ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [15,13,14,9,10,11,4,5,6,12,1,2,3,7,8] => ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [15,13,14,8,9,10,4,5,6,11,1,2,3,7,12] => ? = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> [15,12,13,10,11,14,6,7,8,9,1,2,3,4,5] => ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> [15,12,13,10,11,14,5,6,7,8,1,2,3,4,9] => ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [15,12,13,9,10,14,6,7,8,11,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [15,12,13,9,10,14,5,6,7,11,1,2,3,4,8] => ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [15,12,13,8,9,14,5,6,7,10,1,2,3,4,11] => ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [15,12,13,9,10,14,4,5,6,11,1,2,3,7,8] => ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [15,12,13,8,9,14,4,5,6,10,1,2,3,7,11] => ? = 0 + 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [15,11,12,9,10,13,6,7,8,14,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [15,11,12,9,10,13,5,6,7,14,1,2,3,4,8] => ? = 1 + 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [15,11,12,8,9,13,5,6,7,14,1,2,3,4,10] => ? = 0 + 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [15,11,12,9,10,13,4,5,6,14,1,2,3,7,8] => ? = 1 + 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ? => ? = 0 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [15,12,13,8,9,14,6,7,10,11,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [15,12,13,8,9,14,5,6,10,11,1,2,3,4,7] => ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [15,12,13,7,8,14,5,6,9,10,1,2,3,4,11] => ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [15,12,13,8,9,14,4,5,10,11,1,2,3,6,7] => ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ? => ? = 0 + 3
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [15,11,12,8,9,13,6,7,10,14,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [15,11,12,8,9,13,5,6,10,14,1,2,3,4,7] => ? = 1 + 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? => ? = 0 + 3
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? => ? = 1 + 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [15,11,12,7,8,13,4,5,9,14,1,2,3,6,10] => ? = 0 + 3
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [15,11,12,6,7,13,4,5,8,14,1,2,3,9,10] => ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [15,12,13,8,9,14,3,4,10,11,1,2,5,6,7] => ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [15,12,13,7,8,14,3,4,9,10,1,2,5,6,11] => ? = 0 + 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [15,11,12,8,9,13,3,4,10,14,1,2,5,6,7] => ? = 1 + 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? => ? = 0 + 3
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [15,11,12,6,7,13,3,4,8,14,1,2,5,9,10] => ? = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> [14,13,15,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> [14,13,15,10,11,12,5,6,7,8,1,2,3,4,9] => ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [14,13,15,9,10,11,6,7,8,12,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [14,13,15,9,10,11,5,6,7,12,1,2,3,4,8] => ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [14,13,15,8,9,10,5,6,7,11,1,2,3,4,12] => ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [14,13,15,9,10,11,4,5,6,12,1,2,3,7,8] => ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [14,13,15,8,9,10,4,5,6,11,1,2,3,7,12] => ? = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> [14,12,15,10,11,13,6,7,8,9,1,2,3,4,5] => ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> [14,12,15,10,11,13,5,6,7,8,1,2,3,4,9] => ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [14,12,15,9,10,13,6,7,8,11,1,2,3,4,5] => ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [14,12,15,9,10,13,5,6,7,11,1,2,3,4,8] => ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [14,12,15,8,9,13,5,6,7,10,1,2,3,4,11] => ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [14,12,15,9,10,13,4,5,6,11,1,2,3,7,8] => ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [14,12,15,8,9,13,4,5,6,10,1,2,3,7,11] => ? = 0 + 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,5],[5]]
=> [14,11,15,9,10,12,6,7,8,13,1,2,3,4,5] => ? = 2 + 3
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St001556
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 0
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? = 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ? = 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => ? = 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => ? = 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => ? = 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => ? = 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => ? = 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => ? = 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ? = 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ? = 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ? = 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ? = 0
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ? = 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ? = 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ? = 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ? = 0
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ? = 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ? = 0
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ? = 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ? = 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ? = 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ? = 0
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ? = 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? = 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => ? = 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ? = 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ? = 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ? = 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ? = 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ? = 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => ? = 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => ? = 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => ? = 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,2,5,6] => ? = 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => ? = 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,2,5,6] => ? = 0
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => ? = 2
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001207
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,3,2] => 1 = 0 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => [1,3,2,4] => 1 = 0 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,2,4,3] => 1 = 0 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [1,4,2,3] => 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [1,3,2,4] => 1 = 0 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => [1,4,3,2] => 2 = 1 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [1,2,4,3] => 1 = 0 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,4,3,2] => 2 = 1 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,4,3] => 1 = 0 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [5,4,3,2,1] => [1,5,2,4,3] => ? = 3 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [4,5,3,2,1] => [1,4,2,5,3] => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [5,3,4,2,1] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [3,5,4,2,1] => [1,3,4,2,5] => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [4,3,5,2,1] => [1,4,2,3,5] => ? = 0 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [3,5,4,2,1] => [1,3,4,2,5] => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [3,4,5,2,1] => [1,3,5,2,4] => ? = 0 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [5,4,2,3,1] => [1,5,2,4,3] => ? = 3 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [4,5,2,3,1] => [1,4,3,2,5] => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [5,2,4,3,1] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [4,2,5,3,1] => [1,4,3,5,2] => ? = 0 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,4,5,3,1] => [1,2,4,3,5] => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [5,3,2,4,1] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,5,2,4,1] => [1,3,2,5,4] => ? = 1 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,2,5,4,1] => [1,3,5,2,4] => ? = 0 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,5,2,4,1] => [1,3,2,5,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,2,5,4,1] => [1,3,5,2,4] => ? = 0 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [5,2,4,3,1] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [4,2,5,3,1] => [1,4,3,5,2] => ? = 0 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,4,5,3,1] => [1,2,4,3,5] => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [5,2,3,4,1] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,5,3,4,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,2,5,4,1] => [1,3,5,2,4] => ? = 0 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,5,3,4,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,3,5,4,1] => [1,2,3,5,4] => ? = 0 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,2,5,4,1] => [1,3,5,2,4] => ? = 0 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,4,5,3,1] => [1,2,4,3,5] => ? = 0 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,5,3,4,1] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,3,5,4,1] => [1,2,3,5,4] => ? = 0 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,3,5,4,1] => [1,2,3,5,4] => ? = 0 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [5,4,3,1,2] => [1,5,2,4,3] => ? = 3 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [4,5,3,1,2] => [1,4,2,5,3] => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [5,3,4,1,2] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [3,5,4,1,2] => [1,3,4,2,5] => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [4,3,5,1,2] => [1,4,2,3,5] => ? = 0 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [3,5,4,1,2] => [1,3,4,2,5] => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [3,4,5,1,2] => [1,3,5,2,4] => ? = 0 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [5,4,1,3,2] => [1,5,2,4,3] => ? = 3 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [4,5,1,3,2] => [1,4,3,2,5] => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [5,1,4,3,2] => [1,5,2,3,4] => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [4,1,5,3,2] => [1,4,3,5,2] => ? = 0 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [1,2,5,3,4] => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,4,5,3,2] => [1,2,4,3,5] => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [5,3,1,4,2] => [1,5,2,3,4] => ? = 2 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000133
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2 = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 2 + 2
Description
The "bounce" of a permutation.
Matching statistic: St001085
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => 2 = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10,15] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,4,8,14,2,5,9,15,1,3,6,10,11] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,4,9,13,2,5,10,14,1,3,6,7,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,4,9,14,2,5,10,15,1,3,6,7,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,4,9,15,2,5,10,11,1,3,6,7,12] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,4,10,14,2,5,11,15,1,3,6,7,8] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,4,10,15,2,5,11,12,1,3,6,7,8] => ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,5,8,13,2,6,9,14,1,3,4,10,15] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,5,8,14,2,6,9,15,1,3,4,10,11] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,5,9,13,2,6,10,14,1,3,4,7,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,5,9,14,2,6,10,15,1,3,4,7,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,5,9,15,2,6,10,11,1,3,4,7,12] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,5,10,14,2,6,11,15,1,3,4,7,8] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,5,10,15,2,6,11,12,1,3,4,7,8] => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [11,9,12,5,10,13,2,6,7,14,1,3,4,8,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,9,13,5,10,14,2,6,7,15,1,3,4,8,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,4],[3,4,5],[4,5],[5]]
=> [13,9,14,5,10,15,2,6,7,11,1,3,4,8,12] => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,10,13,5,11,14,2,6,7,15,1,3,4,8,9] => ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ? => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,6,9,13,2,7,10,14,1,3,4,5,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,6,9,14,2,7,10,15,1,3,4,5,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,6,9,15,2,7,10,11,1,3,4,5,12] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,6,10,14,2,7,11,15,1,3,4,5,8] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ? => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [11,9,12,6,10,13,2,7,8,14,1,3,4,5,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,9,13,6,10,14,2,7,8,15,1,3,4,5,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ? => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? => ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> [13,10,14,6,11,15,2,7,8,12,1,3,4,5,9] => ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> [13,11,14,6,12,15,2,7,8,9,1,3,4,5,10] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,7,10,14,2,8,11,15,1,3,4,5,6] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,7,10,15,2,8,11,12,1,3,4,5,6] => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,10,13,7,11,14,2,8,9,15,1,3,4,5,6] => ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ? => ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,3],[3,4,5],[4,5],[5]]
=> [13,11,14,7,12,15,2,8,9,10,1,3,4,5,6] => ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,4,8,13,3,5,9,14,1,2,6,10,15] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,4,8,14,3,5,9,15,1,2,6,10,11] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,4,9,13,3,5,10,14,1,2,6,7,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,4,9,14,3,5,10,15,1,2,6,7,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,4,9,15,3,5,10,11,1,2,6,7,12] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,4,10,14,3,5,11,15,1,2,6,7,8] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,4,10,15,3,5,11,12,1,2,6,7,8] => ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,5,8,13,3,6,9,14,1,2,4,10,15] => ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,5,8,14,3,6,9,15,1,2,4,10,11] => ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,5,9,13,3,6,10,14,1,2,4,7,15] => ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,5,9,14,3,6,10,15,1,2,4,7,11] => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,5,9,15,3,6,10,11,1,2,4,7,12] => ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,5,10,14,3,6,11,15,1,2,4,7,8] => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,5,10,15,3,6,11,12,1,2,4,7,8] => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,3,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [11,9,12,5,10,13,3,6,7,14,1,2,4,8,15] => ? = 2 + 2
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001227
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 + 2
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001803
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,3],[2,4],[5,6],[7,8]]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2],[3,5],[4,6],[7,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> 2 = 0 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2],[3,4],[5,7],[6,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,3],[2,4],[5,7],[6,8]]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2],[3,5],[4,7],[6,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2],[3,6],[4,7],[5,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2],[3,5],[4,7],[6,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2],[3,6],[4,7],[5,8]]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2 = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10]]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10]]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,3],[2,4],[5,7],[6,8],[9,10]]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10]]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10]]
=> ? = 3 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10]]
=> ? = 2 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10]]
=> ? = 2 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ? = 0 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2],[3,6],[4,7],[5,9],[8,10]]
=> ? = 2 + 2
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000115
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000115: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000115: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [[3,2,1],[3,2],[3]]
=> 3 = 0 + 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [[3,2,1],[3,1],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,2],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,2],[3]]
=> 3 = 0 + 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[4]]
=> 4 = 1 + 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[3]]
=> 3 = 0 + 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[3]]
=> 3 = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[5,4],[5]]
=> ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[5,4],[4]]
=> ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[5,3],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[5,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[5,3],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[4,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,3],[4,3],[3]]
=> ? = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,4],[5]]
=> ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,4],[4]]
=> ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,3],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,3],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[4,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[4,3],[3]]
=> ? = 0 + 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,2],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,2],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[5,2],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,4,2],[4,2],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ? = 0 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[5,3],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[5,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[5,3],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[4,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ? = 0 + 3
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[5,2],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[5,2],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[4,2],[3]]
=> ? = 0 + 3
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[5,3,2],[3,2],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[4,3,2],[4,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[4,3,2],[4,3],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[4,3,2],[4,2],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ? = 0 + 3
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> [[5,4,3,2,1],[5,4,3,2],[4,3,2],[3,2],[3]]
=> ? = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[5,4],[5]]
=> ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[5,4],[4]]
=> ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[5,3],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[5,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[5,3],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[4,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,3],[4,3],[3]]
=> ? = 0 + 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[5,4],[5]]
=> ? = 3 + 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[5,4],[4]]
=> ? = 2 + 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[5,3],[5]]
=> ? = 2 + 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[5,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[5,3],[3]]
=> ? = 0 + 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[4,3],[4]]
=> ? = 1 + 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[4,3],[3]]
=> ? = 0 + 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> [[5,4,3,2,1],[5,4,3,1],[5,4,2],[5,2],[5]]
=> ? = 2 + 3
Description
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St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
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