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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St001933
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[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]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000744
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [[1,2,3,7,12],[4,5,6,11,17],[8,9,10,16],[13,14,15]]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3]
=> [[1,2,3,7,16],[4,5,6,11],[8,9,10,15],[12,13,14]]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3]
=> [[1,2,3,10,15],[4,5,6,14],[7,8,9],[11,12,13]]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> [[1,2,3,7,12,18],[4,5,6,11,17],[8,9,10,16],[13,14,15]]
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [[1,2,7,8,13,19],[3,4,11,12,18],[5,6,16,17],[9,10],[14,15]]
=> ? = 5
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [[1,2,5,6,11],[3,4,9,10,16],[7,8,14,15],[12,13]]
=> ? = 4
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 4
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 4
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St001113
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 5 - 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 5 - 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 5 - 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Matching statistic: St001219
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001219: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001219: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 5 - 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 5 - 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 5 - 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
Description
Number 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.
Matching statistic: St000735
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000735: Standard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000735: Standard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 3 + 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 6 = 3 + 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 6 = 3 + 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 2 + 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 2 + 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> ? = 4 + 3
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> ? = 4 + 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> ? = 4 + 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> ? = 3 + 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> ? = 3 + 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> ? = 3 + 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 3 + 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> ? = 3 + 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> ? = 4 + 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> ? = 4 + 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> ? = 4 + 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> ? = 3 + 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> ? = 3 + 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 3 + 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 6 = 3 + 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> ? = 3 + 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> ? = 4 + 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> 7 = 4 + 3
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> ? = 3 + 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 3 + 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 6 = 3 + 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 6 = 3 + 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> ? = 3 + 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> ? = 2 + 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> ? = 2 + 3
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> ? = 2 + 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 2 + 3
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> ? = 2 + 3
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 2 + 3
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 2 + 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> ? = 2 + 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 2 + 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> ? = 3 + 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 3 + 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> ? = 3 + 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> ? = 3 + 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> ? = 2 + 3
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 2 + 3
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 2 + 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15]]
=> ? = 5 + 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 5 + 3
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 5 + 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15,16,17]]
=> ? = 4 + 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13],[14,15,16]]
=> ? = 4 + 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14,15]]
=> ? = 4 + 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> ? = 4 + 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18]]
=> ? = 4 + 3
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17],[18,19]]
=> ? = 5 + 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? = 5 + 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> ? = 5 + 3
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15,16]]
=> ? = 4 + 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 6 = 3 + 3
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> 7 = 4 + 3
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 6 = 3 + 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 6 = 3 + 3
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 6 = 3 + 3
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> 7 = 4 + 3
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 6 = 3 + 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 6 = 3 + 3
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 6 = 3 + 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 5 = 2 + 3
Description
The last entry on the main diagonal of a standard tableau.
Matching statistic: St001330
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(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)
=> ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(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)
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 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]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,8,5,1,6,2,3,4] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [7,8,1,5,6,2,3,4] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [7,8,4,6,1,2,3,5] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,8,6,5,1,7,3,4] => ([(0,4),(1,2),(1,3),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [6,8,5,1,2,7,3,4] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> 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]
=> [5,1,2,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
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: St001685
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => 3 = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => 2 = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => 3 = 4 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => 3 = 4 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,6,4,2] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,5,7,4,2] => 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,4,2] => 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => 2 = 3 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,3,7,6,5,2] => 3 = 4 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,6,3,2] => 3 = 4 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,4,6,5,3,2,7] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => ? = 3 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 3 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,4,6,3,2,7] => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => ? = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,4,3,2,7] => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => ? = 2 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 2 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 3 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => ? = 3 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,7,6,4] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,6,5,4,8] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,6,8,5,4] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,6,5,4] => ? = 4 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => ? = 5 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,2,4,5,8,7,6,3] => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,4,6,8,7,5,3] => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,6,5,8,3] => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,6,8,5,3] => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,2,4,7,8,6,5,3] => ? = 4 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,4,8,7,6,5,3] => ? = 4 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => ? = 5 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,7,4,3] => ? = 5 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,5,7,6,4,3,8] => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,6,4,8,3] => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,5,7,6,8,4,3] => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,6,4,3] => ? = 4 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,5,8,7,6,4,3] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? = 3 - 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000329
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St001192
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
Description
The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A.
Matching statistic: St001210
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
Description
Gives 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.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001481The minimal height of a peak of a Dyck path. St001231The 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001645The pebbling number of a connected graph.
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