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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
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(load all 3 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 0
[[1,2]]
=> [2] => [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[[1,2,3,4]]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[3,4]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4,5]]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,5],[3]]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,5],[4]]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4],[5]]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4,5]]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4,5,6]]
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5,6],[3]]
=> [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4,5],[6]]
=> [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,5,6],[3,4]]
=> [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,6],[3,5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,6],[4,5]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4,5],[3,6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4,6]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4],[5,6]]
=> [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,4,6],[3],[5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,6],[4],[5]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[[1,2,4,5],[3],[6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4],[6]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[[1,2,5],[3,4,6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,4],[3,5,6]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,3,6],[2,5],[4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,6],[3,4],[5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001227
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 86%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 86%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 0
[[1,2]]
=> [2] => [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[[1,2,3,4]]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[3,4]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4,5]]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,5],[3]]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,5],[4]]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4],[5]]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4,5]]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4,5,6]]
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5,6],[3]]
=> [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4,5],[6]]
=> [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,5,6],[3,4]]
=> [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,6],[3,5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,6],[4,5]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4,5],[3,6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4,6]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4],[5,6]]
=> [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,4,6],[3],[5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,6],[4],[5]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[[1,2,4,5],[3],[6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4],[6]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[[1,2,5],[3,4,6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,4],[3,5,6]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,3,6],[2,5],[4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,6],[3,4],[5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,4,5,6,7]]
=> [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,3,4,5,6,7],[2]]
=> [1,6] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[[1,2,4,5,6,7],[3]]
=> [2,5] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[1,2,3,5,6,7],[4]]
=> [3,4] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,2,3,4,6,7],[5]]
=> [4,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,2,3,4,5,7],[6]]
=> [5,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
[[1,2,3,4,5,6],[7]]
=> [6,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,5,6,7],[2,4]]
=> [1,2,4] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,5,6,7],[3,4]]
=> [2,5] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[1,3,4,6,7],[2,5]]
=> [1,3,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[[1,2,4,6,7],[3,5]]
=> [2,2,3] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,2,3,6,7],[4,5]]
=> [3,4] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,2,4,5,7],[3,6]]
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,5,7],[4,6]]
=> [3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[[1,2,3,4,7],[5,6]]
=> [4,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,2,4,5,6],[3,7]]
=> [2,4,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,3,5,6],[4,7]]
=> [3,3,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[1,2,3,4,6],[5,7]]
=> [4,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[1,2,3,4,5],[6,7]]
=> [5,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
[[1,3,5,6,7],[2],[4]]
=> [1,2,4] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[[1,3,4,6,7],[2],[5]]
=> [1,3,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[[1,2,4,6,7],[3],[5]]
=> [2,2,3] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,2,4,5,7],[3],[6]]
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,5,7],[4],[6]]
=> [3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[[1,2,3,4,7],[5],[6]]
=> [4,1,2] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,2,4,5,6],[3],[7]]
=> [2,4,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,3,5,6],[4],[7]]
=> [3,3,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[1,2,3,4,6],[5],[7]]
=> [4,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,3,5,7],[2,4,6]]
=> [1,2,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[1,2,5,7],[3,4,6]]
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,3,4,7],[2,5,6]]
=> [1,3,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[[1,2,4,7],[3,5,6]]
=> [2,2,3] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,2,3,7],[4,5,6]]
=> [3,4] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,2,5,6],[3,4,7]]
=> [2,4,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,4,6],[3,5,7]]
=> [2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,6],[4,5,7]]
=> [3,3,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[1,2,4,5],[3,6,7]]
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,5],[4,6,7]]
=> [3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[[1,2,3,4],[5,6,7]]
=> [4,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,6,7],[3,4],[5]]
=> [2,2,3] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,3,5,7],[2,6],[4]]
=> [1,2,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[1,3,4,7],[2,6],[5]]
=> [1,3,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[[1,2,4,7],[3,6],[5]]
=> [2,2,3] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,3,5,7],[2,4],[6]]
=> [1,2,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[1,2,5,7],[3,4],[6]]
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,7],[4,5],[6]]
=> [3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[[1,2,5,6],[3,7],[4]]
=> [2,1,3,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,4,6],[3,7],[5]]
=> [2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
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