Your data matches 512 different statistics following compositions of up to 3 maps.
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Mp00007: Alternating sign matrices to Dyck pathDyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00103: Dyck paths peeling mapDyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00103: Dyck paths peeling mapDyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
Description
The number of indecomposable three dimensional modules with projective dimension one. It return zero when there are no such modules.
Matching statistic: St000254
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00151: Permutations to cycle typeSet partitions
St000254: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
Description
The nesting number of a set partition. This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000396
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00140: Dyck paths logarithmic height to pruning numberBinary trees
St000396: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> 2
Description
The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St001761: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
Description
The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Mp00007: Alternating sign matrices to Dyck pathDyck paths
Mp00103: Dyck paths peeling mapDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St001006: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 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]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
Description
Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
The following 502 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001256Number of simple reflexive modules that are 2-stable reflexive. St001665The number of pure excedances of a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000649The number of 3-excedences of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001513The number of nested exceedences of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000889The number of alternating sign matrices with the same antidiagonal sums. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001498The normalised height of a Nakayama algebra with magnitude 1. St000326The position of the first one in a binary word after appending a 1 at the end. St000847The number of standard Young tableaux whose descent set is the binary word. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001722The number of minimal chains with small intervals between a binary word and the top element. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St000921The number of internal inversions of a binary word. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000264The girth of a graph, which is not a tree. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000478Another weight of a partition according to Alladi. St000640The rank of the largest boolean interval in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000477The weight of a partition according to Alladi. St000807The sum of the heights of the valleys of the associated bargraph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000741The Colin de Verdière graph invariant. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St000124The cardinality of the preimage of the Simion-Schmidt map. St000405The number of occurrences of the pattern 1324 in a permutation. St000731The number of double exceedences of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001597The Frobenius rank of a skew partition. St001260The permanent of an alternating sign matrix. St000406The number of occurrences of the pattern 3241 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001381The fertility of a permutation. St000805The number of peaks of the associated bargraph. St000787The number of flips required to make a perfect matching noncrossing. St000779The tier of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000666The number of right tethers of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000717The number of ordinal summands of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001850The number of Hecke atoms of a permutation. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001846The number of elements which do not have a complement in the lattice. St000297The number of leading ones in a binary word. St000288The number of ones in a binary word. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St000842The breadth of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000189The number of elements in the poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001964The interval resolution global dimension of a poset. St001651The Frankl number of a lattice. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001130The number of two successive successions in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St000891The number of distinct diagonal sums of a permutation matrix. St000862The number of parts of the shifted shape of a permutation. St000534The number of 2-rises of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000664The number of right ropes of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000487The length of the shortest cycle of a permutation. St000210Minimum over maximum difference of elements in cycles. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000035The number of left outer peaks of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000764The number of strong records in an integer composition. St001618The cardinality of the Frattini sublattice of a lattice. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St000768The number of peaks in an integer composition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001833The number of linear intervals in a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001645The pebbling number of a connected graph. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000352The Elizalde-Pak rank of a permutation. St000402Half the size of the symmetry class of a permutation. St000651The maximal size of a rise in a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000696The number of cycles in the breakpoint graph of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St000058The order of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001589The nesting number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000407The number of occurrences of the pattern 2143 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000062The length of the longest increasing subsequence of the permutation. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000733The row containing the largest entry of a standard tableau. St000822The Hadwiger number of the graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001060The distinguishing index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001469The holeyness of a permutation. St001481The minimal height of a peak of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. St000037The sign of a permutation. St000061The number of nodes on the left branch of a binary tree. St000069The number of maximal elements of a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000084The number of subtrees. St000096The number of spanning trees of a graph. St000155The number of exceedances (also excedences) of a permutation. St000163The size of the orbit of the set partition under rotation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000230Sum of the minimal elements of the blocks of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000450The number of edges minus the number of vertices plus 2 of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000627The exponent of a binary word. St000635The number of strictly order preserving maps of a poset into itself. St000654The first descent of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000756The sum of the positions of the left to right maxima of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000843The decomposition number of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000958The number of Bruhat factorizations of a permutation. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001117The game chromatic index of a graph. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001220The width of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001518The number of graphs with the same ordinary spectrum as the given graph. St001568The smallest positive integer that does not appear twice in the partition. St001729The number of visible descents of a permutation. St001734The lettericity of a graph. St001737The number of descents of type 2 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001828The Euler characteristic of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000002The number of occurrences of the pattern 123 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000095The number of triangles of a graph. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000347The inversion sum of a binary word. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St000650The number of 3-rises of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000895The number of ones on the main diagonal of an alternating sign matrix. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001520The number of strict 3-descents. St001536The number of cyclic misalignments of a permutation. St001557The number of inversions of the second entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001895The oddness of a signed permutation. St001948The number of augmented double ascents of a permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001555The order of a signed permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001596The number of two-by-two squares inside a skew partition. St000052The number of valleys of a Dyck path not on the x-axis. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001545The second Elser number of a connected graph. St001587Half of the largest even part of an integer partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001861The number of Bruhat lower covers of a permutation. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000068The number of minimal elements in a poset. St000160The multiplicity of the smallest part of a partition. St000667The greatest common divisor of the parts of the partition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001696The natural major index of a standard Young tableau. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001769The reflection length of a signed permutation. St001851The number of Hecke atoms of a signed permutation. St000237The number of small exceedances. St000001The number of reduced words for a permutation. St000883The number of longest increasing subsequences of a permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001852The size of the conjugacy class of the signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St001848The atomic length of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001372The length of a longest cyclic run of ones of a binary word. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000383The last part of an integer composition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000647The number of big descents of a permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000662The staircase size of the code of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000703The number of deficiencies of a permutation. St000871The number of very big ascents of a permutation. St000463The number of admissible inversions of a permutation. St000884The number of isolated descents of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001684The reduced word complexity of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000246The number of non-inversions of a permutation.