Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001584
Mp00028: Dyck paths reverseDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001584: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6
Description
The area statistic between a Dyck path and its bounce path. The bounce path [[Mp00099]] is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.
St001232: Dyck paths ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> ? = 4
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> ? = 4
[1,1,0,1,1,0,1,0,0,0]
=> ? = 5
[1,1,0,1,1,1,0,0,0,0]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> ? = 4
[1,1,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,1,0,1,1,0,0,0,0]
=> 6
[1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> 6
[1,0,1,1,1,1,0,0,1,0,0,0]
=> 7
[1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[1,1,0,1,1,1,1,0,0,0,0,0]
=> 8
[1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.