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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
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(load all 15 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 7
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 10
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 8
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 7
[3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 9
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 8
[4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 7
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 11
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001489
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 42%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 42%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [5,4,2,1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [6,3,5,2,1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,3,4,5,6,1] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [7,5,2,1,3,6,4] => ? = 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,7,2,6,4,1,3] => ? = 5
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [7,3,4,6,2,1,5] => ? = 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,1,2,3,5,6,7,4] => ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [8,3,6,2,1,4,7,5] => ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,7,2] => ? = 3
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [6,8,5,3,7,2,1,4] => ? = 7
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [8,3,4,5,7,2,1,6] => ? = 6
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [9,6,2,1,3,4,7,8,5] => ? = 3
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [6,1,2,9,4,5,7,8,3] => ? = 4
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [9,3,4,7,2,1,5,8,6] => ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [5,7,2,9,4,8,6,1,3] => ? = 5
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [7,9,4,6,3,8,2,1,5] => ? = 9
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [10,1,2,3,4,6,7,8,9,5] => ? = 2
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [10,3,7,2,1,4,5,8,9,6] => ? = 4
[3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4] => [7,10,2,1,8,5,3,6,9,4] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,10,7,9,2] => ? = 3
[2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> [9,7,5,10,3,8,2,6,1,4] => [6,8,10,3,7,5,9,2,1,4] => ? = 7
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? => ? = 7
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? => ? = 10
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => [12,1,2,3,4,5,7,8,9,10,11,6] => ? = 2
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? => ? = 4
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? => ? = 5
[3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6] => ? => ? = 8
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,10,7,12,9,11,2] => ? = 3
[4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? => ? => ? = 7
[3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? => ? => ? = 9
[3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? => ? = 8
[4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? => ? => ? = 7
[3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? => ? = 11
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? => ? => ? = 4
[3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? => ? = 5
[4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ? => ? = 6
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St001330
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 58%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10 + 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,5,3,4,2,7,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9 + 1
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [7,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11 + 1
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000354
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 42%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 42%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [5,4,2,1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [6,3,5,2,1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,3,4,5,6,1] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [7,5,2,1,3,6,4] => ? = 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,7,2,6,4,1,3] => ? = 5
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [7,3,4,6,2,1,5] => ? = 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,1,2,3,5,6,7,4] => ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [8,3,6,2,1,4,7,5] => ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,7,2] => ? = 3
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [6,8,5,3,7,2,1,4] => ? = 7
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [8,3,4,5,7,2,1,6] => ? = 6
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [9,6,2,1,3,4,7,8,5] => ? = 3
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [6,1,2,9,4,5,7,8,3] => ? = 4
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [9,3,4,7,2,1,5,8,6] => ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [5,7,2,9,4,8,6,1,3] => ? = 5
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [7,9,4,6,3,8,2,1,5] => ? = 9
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [10,1,2,3,4,6,7,8,9,5] => ? = 2
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [10,3,7,2,1,4,5,8,9,6] => ? = 4
[3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4] => [7,10,2,1,8,5,3,6,9,4] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,10,7,9,2] => ? = 3
[2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> [9,7,5,10,3,8,2,6,1,4] => [6,8,10,3,7,5,9,2,1,4] => ? = 7
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? => ? = 7
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? => ? = 10
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => [12,1,2,3,4,5,7,8,9,10,11,6] => ? = 2
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? => ? = 4
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? => ? = 5
[3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6] => ? => ? = 8
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,10,7,12,9,11,2] => ? = 3
[4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? => ? => ? = 7
[3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? => ? => ? = 9
[3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? => ? = 8
[4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? => ? => ? = 7
[3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? => ? = 11
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? => ? => ? = 4
[3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? => ? = 5
[4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ? => ? = 6
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000829
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 42%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 42%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [5,4,2,1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [6,3,5,2,1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,3,4,5,6,1] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [7,5,2,1,3,6,4] => ? = 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,7,2,6,4,1,3] => ? = 5
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [7,3,4,6,2,1,5] => ? = 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,1,2,3,5,6,7,4] => ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [8,3,6,2,1,4,7,5] => ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,7,2] => ? = 3
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [6,8,5,3,7,2,1,4] => ? = 7
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [8,3,4,5,7,2,1,6] => ? = 6
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [9,6,2,1,3,4,7,8,5] => ? = 3
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [6,1,2,9,4,5,7,8,3] => ? = 4
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [9,3,4,7,2,1,5,8,6] => ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [5,7,2,9,4,8,6,1,3] => ? = 5
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [7,9,4,6,3,8,2,1,5] => ? = 9
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [10,1,2,3,4,6,7,8,9,5] => ? = 2
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [10,3,7,2,1,4,5,8,9,6] => ? = 4
[3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4] => [7,10,2,1,8,5,3,6,9,4] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,10,7,9,2] => ? = 3
[2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> [9,7,5,10,3,8,2,6,1,4] => [6,8,10,3,7,5,9,2,1,4] => ? = 7
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? => ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? => ? = 7
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? => ? = 10
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => [12,1,2,3,4,5,7,8,9,10,11,6] => ? = 2
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? => ? = 4
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? => ? = 5
[3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6] => ? => ? = 8
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,10,7,12,9,11,2] => ? = 3
[4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? => ? => ? = 7
[3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? => ? => ? = 9
[3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? => ? = 8
[4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? => ? => ? = 7
[3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? => ? = 11
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? => ? => ? = 4
[3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? => ? = 5
[4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ? => ? = 6
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St001207
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [1] => [1] => ? = 0
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1,4] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => ? = 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,5,2,1] => ? = 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => ? = 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => ? = 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => [3,2,4,5,1,6] => ? = 4
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,3,2,1,5] => ? = 3
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => [4,3,5,6,2,1] => ? = 7
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => ? = 6
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,2,6,1] => [3,2,4,1,5,6] => ? = 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => [3,2,4,5,6,1,7] => ? = 5
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,3,2,6,1] => [4,3,5,2,1,6] => ? = 5
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => [4,3,5,6,7,2,1] => ? = 9
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [3,2,1,4,5,6] => ? = 2
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => [3,2,4,5,1,6,7] => ? = 4
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [5,4,6,3,2,1] => ? = 7
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,3,2,1] => [4,3,2,1,5,6] => ? = 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => [4,3,5,6,2,1,7] => ? = 7
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => [3,2,4,1,5,6,7] => ? = 3
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [5,4,3,2,6,1] => ? = 7
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => [5,4,6,7,3,2,1] => ? = 10
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [4,3,5,2,1,6,7] => ? = 5
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [3,2,1,4,5,6,7] => ? = 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => [5,4,3,2,1,6] => ? = 4
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 5
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => [5,4,6,3,2,7,1] => ? = 8
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 3
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => [5,4,6,3,2,1,7] => ? = 7
[3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,7,4,3,2,1] => ? = 9
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => [5,4,3,2,6,7,1] => ? = 8
[4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => [5,4,3,2,6,1,7] => ? = 7
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,3,7,2,1] => ? = 11
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => [5,4,3,2,1,6,7] => ? = 4
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,5,4,3,2,1,7] => ? = 5
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000366
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> [4,6,7,3,9,5,2,11,8,12,10,1] => ? = 4
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 3
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
=> [5,6,7,9,4,3,2,11,8,12,10,1] => ? = 7
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
=> [4,5,7,3,2,9,6,11,8,13,10,14,12,1] => ? = 6
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
=> [4,6,8,3,9,5,11,7,2,12,10,1] => ? = 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 4
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,14),(2,7),(3,4),(5,6),(8,9),(10,11),(12,13)]
=> [4,6,7,3,9,5,2,11,8,13,10,14,12,1] => ? = 5
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
=> [5,7,8,9,4,11,6,3,2,12,10,1] => ? = 5
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [(1,14),(2,7),(3,6),(4,5),(8,9),(10,11),(12,13)]
=> [5,6,7,9,4,3,2,11,8,13,10,14,12,1] => ? = 9
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 2
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> [4,6,8,3,9,5,11,7,2,13,10,14,12,1] => ? = 4
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
=> [6,7,8,9,11,5,4,3,2,12,10,1] => ? = 7
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)]
=> [5,7,9,10,4,11,6,12,8,3,2,1] => ? = 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,5),(6,7),(10,11),(12,13)]
=> [5,7,8,9,4,11,6,3,2,13,10,14,12,1] => ? = 7
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,6),(7,8),(9,10),(12,13)]
=> [4,6,8,3,10,5,11,7,13,9,2,14,12,1] => ? = 3
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 7
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,7),(5,6),(10,11),(12,13)]
=> [6,7,8,9,11,5,4,3,2,13,10,14,12,1] => ? = 10
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,5),(6,7),(8,9),(12,13)]
=> [5,7,9,10,4,11,6,13,8,3,2,14,12,1] => ? = 5
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [4,6,8,3,10,5,12,7,13,9,14,11,2,1] => ? = 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)]
=> [6,8,9,10,11,5,12,7,4,3,2,1] => ? = 4
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 5
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,7),(5,6),(8,9),(12,13)]
=> [6,7,9,10,11,5,4,13,8,3,2,14,12,1] => ? = 8
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [5,7,9,11,4,12,6,13,8,14,10,3,2,1] => ? = 3
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,6),(7,8),(12,13)]
=> [6,8,9,10,11,5,13,7,4,3,2,14,12,1] => ? = 7
[3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,8),(6,7),(12,13)]
=> [7,8,9,10,11,13,6,5,4,3,2,14,12,1] => ? = 9
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [6,7,9,11,12,5,4,13,8,14,10,3,2,1] => ? = 8
[4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [6,8,9,11,12,5,13,7,4,14,10,3,2,1] => ? = 7
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [7,8,9,11,12,13,6,5,4,14,10,3,2,1] => ? = 11
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [6,8,10,11,12,5,13,7,14,9,4,3,2,1] => ? = 4
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1] => ? = 5
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 6
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000352
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 3 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 3 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 5 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 5 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> [4,6,7,3,9,5,2,11,8,12,10,1] => ? = 4 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 3 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
=> [5,6,7,9,4,3,2,11,8,12,10,1] => ? = 7 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
=> [4,5,7,3,2,9,6,11,8,13,10,14,12,1] => ? = 6 + 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
=> [4,6,8,3,9,5,11,7,2,12,10,1] => ? = 3 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 4 + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,14),(2,7),(3,4),(5,6),(8,9),(10,11),(12,13)]
=> [4,6,7,3,9,5,2,11,8,13,10,14,12,1] => ? = 5 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
=> [5,7,8,9,4,11,6,3,2,12,10,1] => ? = 5 + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [(1,14),(2,7),(3,6),(4,5),(8,9),(10,11),(12,13)]
=> [5,6,7,9,4,3,2,11,8,13,10,14,12,1] => ? = 9 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 2 + 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> [4,6,8,3,9,5,11,7,2,13,10,14,12,1] => ? = 4 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
=> [6,7,8,9,11,5,4,3,2,12,10,1] => ? = 7 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)]
=> [5,7,9,10,4,11,6,12,8,3,2,1] => ? = 3 + 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,5),(6,7),(10,11),(12,13)]
=> [5,7,8,9,4,11,6,3,2,13,10,14,12,1] => ? = 7 + 1
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,6),(7,8),(9,10),(12,13)]
=> [4,6,8,3,10,5,11,7,13,9,2,14,12,1] => ? = 3 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 7 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,7),(5,6),(10,11),(12,13)]
=> [6,7,8,9,11,5,4,3,2,13,10,14,12,1] => ? = 10 + 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,5),(6,7),(8,9),(12,13)]
=> [5,7,9,10,4,11,6,13,8,3,2,14,12,1] => ? = 5 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [4,6,8,3,10,5,12,7,13,9,14,11,2,1] => ? = 2 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)]
=> [6,8,9,10,11,5,12,7,4,3,2,1] => ? = 4 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 5 + 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,7),(5,6),(8,9),(12,13)]
=> [6,7,9,10,11,5,4,13,8,3,2,14,12,1] => ? = 8 + 1
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [5,7,9,11,4,12,6,13,8,14,10,3,2,1] => ? = 3 + 1
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,6),(7,8),(12,13)]
=> [6,8,9,10,11,5,13,7,4,3,2,14,12,1] => ? = 7 + 1
[3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,8),(6,7),(12,13)]
=> [7,8,9,10,11,13,6,5,4,3,2,14,12,1] => ? = 9 + 1
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [6,7,9,11,12,5,4,13,8,14,10,3,2,1] => ? = 8 + 1
[4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [6,8,9,11,12,5,13,7,4,14,10,3,2,1] => ? = 7 + 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [7,8,9,11,12,13,6,5,4,14,10,3,2,1] => ? = 11 + 1
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [6,8,10,11,12,5,13,7,14,9,4,3,2,1] => ? = 4 + 1
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1] => ? = 5 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 6 + 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St001645
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Values
[1]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[2]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[1,1]
=> 11 => [2] => ([],2)
=> ? = 1 + 1
[3]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1]
=> 111 => [3] => ([],3)
=> ? = 1 + 1
[4]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[2,2]
=> 00 => [2] => ([],2)
=> ? = 2 + 1
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? = 1 + 1
[5]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? = 1 + 1
[6]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[3,3]
=> 11 => [2] => ([],2)
=> ? = 2 + 1
[2,2,2]
=> 000 => [3] => ([],3)
=> ? = 3 + 1
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? = 1 + 1
[7]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[3,3,1]
=> 111 => [3] => ([],3)
=> ? = 3 + 1
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 5 + 1
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 5 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? = 1 + 1
[4,4]
=> 00 => [2] => ([],2)
=> ? = 2 + 1
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? = 4 + 1
[2,2,2,2]
=> 0000 => [4] => ([],4)
=> ? = 3 + 1
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 7 + 1
[2,2,1,1,1,1]
=> 001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? = 6 + 1
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[3,3,3]
=> 111 => [3] => ([],3)
=> ? = 4 + 1
[3,3,1,1,1]
=> 11111 => [5] => ([],5)
=> ? = 5 + 1
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 5 + 1
[2,2,2,1,1,1]
=> 000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 9 + 1
[5,5]
=> 11 => [2] => ([],2)
=> ? = 2 + 1
[4,4,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[3,3,3,1]
=> 1111 => [4] => ([],4)
=> ? = 7 + 1
[2,2,2,2,2]
=> 00000 => [5] => ([],5)
=> ? = 3 + 1
[2,2,2,2,1,1]
=> 000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 7 + 1
[5,5,1]
=> 111 => [3] => ([],3)
=> ? = 3 + 1
[3,3,3,2]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 7 + 1
[3,3,3,1,1]
=> 11111 => [5] => ([],5)
=> ? = 10 + 1
[2,2,2,2,2,1]
=> 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 5 + 1
[6,6]
=> 00 => [2] => ([],2)
=> ? = 2 + 1
[4,4,4]
=> 000 => [3] => ([],3)
=> ? = 4 + 1
[3,3,3,3]
=> 1111 => [4] => ([],4)
=> ? = 5 + 1
[3,3,3,2,1]
=> 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 8 + 1
[2,2,2,2,2,2]
=> 000000 => [6] => ([],6)
=> ? = 3 + 1
[4,4,4,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 7 + 1
[3,3,3,3,1]
=> 11111 => [5] => ([],5)
=> ? = 9 + 1
[3,3,3,2,2]
=> 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 8 + 1
[4,4,4,2]
=> 0000 => [4] => ([],4)
=> ? = 7 + 1
[3,3,3,3,2]
=> 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 11 + 1
[5,5,5]
=> 111 => [3] => ([],3)
=> ? = 4 + 1
[3,3,3,3,3]
=> 11111 => [5] => ([],5)
=> ? = 5 + 1
[4,4,4,4]
=> 0000 => [4] => ([],4)
=> ? = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000007
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 2 = 0 + 2
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 0 + 2
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 3 = 1 + 2
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 3 = 1 + 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 4 = 2 + 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 2 = 0 + 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3 + 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1 + 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 2 = 0 + 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 + 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 3 + 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 4 + 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1 + 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0 + 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 3 + 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 5 + 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 5 + 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1 + 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 + 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> [4,6,7,3,9,5,2,11,8,12,10,1] => ? = 4 + 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 3 + 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
=> [5,6,7,9,4,3,2,11,8,12,10,1] => ? = 7 + 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
=> [4,5,7,3,2,9,6,11,8,13,10,14,12,1] => ? = 6 + 2
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
=> [4,6,8,3,9,5,11,7,2,12,10,1] => ? = 3 + 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 4 + 2
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,14),(2,7),(3,4),(5,6),(8,9),(10,11),(12,13)]
=> [4,6,7,3,9,5,2,11,8,13,10,14,12,1] => ? = 5 + 2
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
=> [5,7,8,9,4,11,6,3,2,12,10,1] => ? = 5 + 2
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [(1,14),(2,7),(3,6),(4,5),(8,9),(10,11),(12,13)]
=> [5,6,7,9,4,3,2,11,8,13,10,14,12,1] => ? = 9 + 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 2 + 2
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> [4,6,8,3,9,5,11,7,2,13,10,14,12,1] => ? = 4 + 2
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
=> [6,7,8,9,11,5,4,3,2,12,10,1] => ? = 7 + 2
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)]
=> [5,7,9,10,4,11,6,12,8,3,2,1] => ? = 3 + 2
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,5),(6,7),(10,11),(12,13)]
=> [5,7,8,9,4,11,6,3,2,13,10,14,12,1] => ? = 7 + 2
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,6),(7,8),(9,10),(12,13)]
=> [4,6,8,3,10,5,11,7,13,9,2,14,12,1] => ? = 3 + 2
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 7 + 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,7),(5,6),(10,11),(12,13)]
=> [6,7,8,9,11,5,4,3,2,13,10,14,12,1] => ? = 10 + 2
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,5),(6,7),(8,9),(12,13)]
=> [5,7,9,10,4,11,6,13,8,3,2,14,12,1] => ? = 5 + 2
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [4,6,8,3,10,5,12,7,13,9,14,11,2,1] => ? = 2 + 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)]
=> [6,8,9,10,11,5,12,7,4,3,2,1] => ? = 4 + 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 5 + 2
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,7),(5,6),(8,9),(12,13)]
=> [6,7,9,10,11,5,4,13,8,3,2,14,12,1] => ? = 8 + 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [5,7,9,11,4,12,6,13,8,14,10,3,2,1] => ? = 3 + 2
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,6),(7,8),(12,13)]
=> [6,8,9,10,11,5,13,7,4,3,2,14,12,1] => ? = 7 + 2
[3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,8),(6,7),(12,13)]
=> [7,8,9,10,11,13,6,5,4,3,2,14,12,1] => ? = 9 + 2
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [6,7,9,11,12,5,4,13,8,14,10,3,2,1] => ? = 8 + 2
[4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [6,8,9,11,12,5,13,7,4,14,10,3,2,1] => ? = 7 + 2
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [7,8,9,11,12,13,6,5,4,14,10,3,2,1] => ? = 11 + 2
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [6,8,10,11,12,5,13,7,14,9,4,3,2,1] => ? = 4 + 2
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1] => ? = 5 + 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 6 + 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000455The second largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000174The flush statistic of a semistandard tableau. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000254The nesting number of a set partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000654The first descent of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree.
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