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Your data matches 54 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1
([],2)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [1,2] => 2
([(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([],4)
=> [4] => 4
([(2,3)],4)
=> [1,3] => 3
([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 2
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
([],5)
=> [5] => 5
([(3,4)],5)
=> [1,4] => 4
([(2,4),(3,4)],5)
=> [1,1,3] => 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 3
([(1,4),(2,3)],5)
=> [2,3] => 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> 1
([],2)
=> [2] => [2]
=> 2
([(0,1)],2)
=> [1,1] => [1,1]
=> 1
([],3)
=> [3] => [3]
=> 3
([(1,2)],3)
=> [1,2] => [2,1]
=> 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> 2
([],4)
=> [4] => [4]
=> 4
([(2,3)],4)
=> [1,3] => [3,1]
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,1,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 3
([],5)
=> [5] => [5]
=> 5
([(3,4)],5)
=> [1,4] => [4,1]
=> 4
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [3,1,1]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,1,1,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,1,1,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> [1]
=> 1
([],2)
=> [2] => [2]
=> [1,1]
=> 2
([(0,1)],2)
=> [1,1] => [1,1]
=> [2]
=> 1
([],3)
=> [3] => [3]
=> [1,1,1]
=> 3
([(1,2)],3)
=> [1,2] => [2,1]
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> [3]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> [2,1]
=> 2
([],4)
=> [4] => [4]
=> [1,1,1,1]
=> 4
([(2,3)],4)
=> [1,3] => [3,1]
=> [2,1,1]
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1]
=> [3,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [2,1,1]
=> 3
([],5)
=> [5] => [5]
=> [1,1,1,1,1]
=> 5
([(3,4)],5)
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 4
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2]
=> [2,2,1]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2]
=> [2,2,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [2,2,1]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
Description
The length of the partition.
Matching statistic: St000734
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> [[1]]
=> 1
([],2)
=> [2] => [2]
=> [[1,2]]
=> 2
([(0,1)],2)
=> [1,1] => [1,1]
=> [[1],[2]]
=> 1
([],3)
=> [3] => [3]
=> [[1,2,3]]
=> 3
([(1,2)],3)
=> [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
([],4)
=> [4] => [4]
=> [[1,2,3,4]]
=> 4
([(2,3)],4)
=> [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
([],5)
=> [5] => [5]
=> [[1,2,3,4,5]]
=> 5
([(3,4)],5)
=> [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 => 0 => 0 = 1 - 1
([],2)
=> [2] => 10 => 01 => 1 = 2 - 1
([(0,1)],2)
=> [1,1] => 11 => 00 => 0 = 1 - 1
([],3)
=> [3] => 100 => 011 => 2 = 3 - 1
([(1,2)],3)
=> [1,2] => 110 => 001 => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => 111 => 000 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 101 => 010 => 1 = 2 - 1
([],4)
=> [4] => 1000 => 0111 => 3 = 4 - 1
([(2,3)],4)
=> [1,3] => 1100 => 0011 => 2 = 3 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => 1110 => 0001 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
([],5)
=> [5] => 10000 => 01111 => 4 = 5 - 1
([(3,4)],5)
=> [1,4] => 11000 => 00111 => 3 = 4 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => 11100 => 00011 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 11010 => 00101 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 11001 => 00110 => 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 11010 => 00101 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [1,1,1,1,1,1,1,1,1,1,1,1] => 111111111111 => 000000000000 => ? = 1 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000983
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 => 1 => 1
([],2)
=> [2] => 10 => 01 => 2
([(0,1)],2)
=> [1,1] => 11 => 11 => 1
([],3)
=> [3] => 100 => 101 => 3
([(1,2)],3)
=> [1,2] => 110 => 011 => 2
([(0,2),(1,2)],3)
=> [1,1,1] => 111 => 111 => 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 101 => 001 => 2
([],4)
=> [4] => 1000 => 0101 => 4
([(2,3)],4)
=> [1,3] => 1100 => 1011 => 3
([(1,3),(2,3)],4)
=> [1,1,2] => 1110 => 0111 => 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1101 => 0011 => 2
([(0,3),(1,2)],4)
=> [2,2] => 1010 => 1001 => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 1111 => 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 1010 => 1001 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 1111 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1101 => 0011 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1011 => 0001 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1001 => 1101 => 3
([],5)
=> [5] => 10000 => 10101 => 5
([(3,4)],5)
=> [1,4] => 11000 => 01011 => 4
([(2,4),(3,4)],5)
=> [1,1,3] => 11100 => 10111 => 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 11010 => 10011 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 11001 => 11011 => 3
([(1,4),(2,3)],5)
=> [2,3] => 10100 => 01001 => 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => 01111 => 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => 01111 => 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => 10100 => 01001 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => 01111 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 11101 => 00111 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 11010 => 10011 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 10110 => 10001 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 11101 => 00111 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 11001 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 10110 => 10001 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => 00011 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 10101 => 11001 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 10010 => 01101 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => 00011 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 10111 => 00001 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 11001 => 2
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,1),(0,5),(0,6),(0,8),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,2),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(2,7),(2,8),(3,4),(3,5),(3,7),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [1,1,1,1,1,1,1,1,1,1,1,1] => 111111111111 => 111111111111 => ? = 1
([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,5),(1,2),(1,9),(2,6),(3,7),(3,9),(4,6),(4,8),(5,7),(6,9),(7,8),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,9),(1,3),(1,8),(2,8),(2,9),(3,5),(4,7),(4,9),(5,7),(5,8),(6,7),(6,8),(6,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,8),(0,9),(1,5),(1,7),(1,9),(2,4),(2,7),(2,8),(3,4),(3,5),(3,7),(4,6),(4,9),(5,6),(5,8),(6,8),(6,9),(7,8),(7,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,7),(0,9),(1,6),(1,8),(2,3),(2,4),(2,5),(3,8),(3,9),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,9),(7,8),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,8),(0,9),(1,6),(1,7),(1,9),(2,5),(2,7),(2,9),(3,5),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,9),(6,9),(7,8),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,7),(0,8),(0,9),(1,2),(1,5),(1,6),(1,9),(2,3),(2,4),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,9),(5,7),(5,8),(6,7),(6,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,5),(0,6),(0,9),(1,4),(1,8),(1,9),(2,3),(2,7),(2,9),(3,4),(3,5),(3,8),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,5),(0,7),(0,9),(1,4),(1,6),(1,9),(2,3),(2,8),(2,9),(3,4),(3,5),(3,9),(4,7),(4,8),(5,6),(5,8),(6,8),(6,9),(7,8),(7,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
([(0,3),(0,8),(0,9),(1,2),(1,7),(1,9),(2,5),(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form 010... or of the form 101....
Matching statistic: St000444
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 95%●distinct values known / distinct values provided: 88%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 95%●distinct values known / distinct values provided: 88%
Values
([],1)
=> [1] => [1] => [1,0]
=> ? = 1
([],2)
=> [2] => [2] => [1,1,0,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> [1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> [1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,1,2,1,1] => [1,1,2,1,2,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> [1,3,3,1,1] => [1,1,3,3,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,2,2,1,1] => [1,1,2,2,1,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,1),(0,5),(0,6),(0,8),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,1),(0,5),(0,7),(0,8),(1,4),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,2,2,1] => [1,2,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> [4,4,1] => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
([(0,4),(0,5),(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(3,6),(3,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,2,1,1,2,1,1] => [1,1,2,1,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,2),(0,3),(1,4),(1,5),(1,6),(1,7),(2,6),(2,7),(2,8),(3,4),(3,5),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,7),(4,8),(5,7),(6,8),(7,8)],9)
=> [3,2,2,1,1] => [1,1,2,2,3] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,2),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(2,7),(2,8),(3,4),(3,5),(3,7),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
([(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)
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
([(3,7),(4,7),(5,7),(6,7)],8)
=> [1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
([],8)
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
([(4,7),(5,6)],8)
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
([(4,7),(5,6),(6,7)],8)
=> [1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
([(4,6),(4,7),(5,6),(5,7)],8)
=> [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 95%●distinct values known / distinct values provided: 88%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 95%●distinct values known / distinct values provided: 88%
Values
([],1)
=> [1] => [1] => [1,0]
=> ? = 1 - 1
([],2)
=> [2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
([(1,2)],3)
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
([(2,3)],4)
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
([(3,4)],5)
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> [1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> [1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,1,2,1,1] => [1,1,2,1,2,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> [1,3,3,1,1] => [1,1,3,3,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,2,2,1,1] => [1,1,2,2,1,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(0,8),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
([(0,1),(0,5),(0,7),(0,8),(1,4),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,2,2,1] => [1,2,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> [4,4,1] => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(3,6),(3,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,2,1,1,2,1,1] => [1,1,2,1,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,2),(0,3),(1,4),(1,5),(1,6),(1,7),(2,6),(2,7),(2,8),(3,4),(3,5),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,7),(4,8),(5,7),(6,8),(7,8)],9)
=> [3,2,2,1,1] => [1,1,2,2,3] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(0,2),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(2,7),(2,8),(3,4),(3,5),(3,7),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
([(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)
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
([(3,7),(4,7),(5,7),(6,7)],8)
=> [1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
([],8)
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
([(4,7),(5,6)],8)
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
([(4,7),(5,6),(6,7)],8)
=> [1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
([(4,6),(4,7),(5,6),(5,7)],8)
=> [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000521
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 88% ●values known / values provided: 94%●distinct values known / distinct values provided: 88%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 88% ●values known / values provided: 94%●distinct values known / distinct values provided: 88%
Values
([],1)
=> [1] => [1,0]
=> [[]]
=> 2 = 1 + 1
([],2)
=> [2] => [1,1,0,0]
=> [[[]]]
=> 3 = 2 + 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [[],[]]
=> 2 = 1 + 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 4 = 3 + 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 3 = 2 + 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 5 = 4 + 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4 = 3 + 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 6 = 5 + 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[],[],[],[[]],[],[],[]]
=> ? = 2 + 1
([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[],[],[],[[]],[],[],[]]
=> ? = 2 + 1
([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,1,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ?
=> ? = 2 + 1
([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[[[]]],[],[]]
=> ? = 3 + 1
([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> [1,3,3,1,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[],[[[]]],[[[]]],[],[]]
=> ? = 3 + 1
([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[[]],[],[],[],[],[]]
=> ? = 2 + 1
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,2,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ?
=> ? = 2 + 1
([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[[]],[],[],[],[],[]]
=> ? = 2 + 1
([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,1),(0,5),(0,6),(0,8),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[[]],[],[],[],[]]
=> ? = 2 + 1
([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[[]],[],[],[],[],[]]
=> ? = 2 + 1
([(0,1),(0,5),(0,7),(0,8),(1,4),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[[]],[[]],[]]
=> ? = 2 + 1
([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[[[[]]]],[]]
=> ? = 4 + 1
([(0,4),(0,5),(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,1,2,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[[]],[],[],[]]
=> ? = 2 + 1
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(3,6),(3,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,2,1,1,2,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[],[[]],[],[]]
=> ? = 2 + 1
([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,2),(0,3),(1,4),(1,5),(1,6),(1,7),(2,6),(2,7),(2,8),(3,4),(3,5),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[[]],[],[],[],[],[]]
=> ? = 2 + 1
([(0,4),(0,5),(0,8),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,7),(4,8),(5,7),(6,8),(7,8)],9)
=> [3,2,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ?
=> ? = 3 + 1
([(0,2),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(2,7),(2,8),(3,4),(3,5),(3,7),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[]]]]]]],[]]
=> ? = 7 + 1
([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 6 + 1
([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[[[]]],[],[]]
=> ? = 3 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 5 + 1
([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[[]]]],[[]],[],[]]
=> ? = 4 + 1
([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[]]],[[]],[],[[]]]
=> ? = 3 + 1
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[]]],[[]],[[]],[]]
=> ? = 3 + 1
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 5 + 1
([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[[]]]],[],[],[[]]]
=> ? = 4 + 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[[]]]],[],[[]],[]]
=> ? = 4 + 1
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[]]],[],[[[]]],[]]
=> ? = 3 + 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[[]]]]]],[],[]]
=> ? = 6 + 1
([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[[]]]]],[],[[]]]
=> ? = 5 + 1
([(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),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[]]],[[]],[],[],[]]
=> ? = 3 + 1
([(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),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[]]]],[],[],[],[]]
=> ? = 4 + 1
([(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),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[[]]]]],[[]],[]]
=> ? = 5 + 1
Description
The number of distinct subtrees of an ordered tree.
A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree [[[[]],[]],[[],[[]]]] on nine nodes has five distinct subtrees.
Matching statistic: St000013
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => [1,0]
=> 1
([],2)
=> [2] => [2] => [1,1,0,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 3
([(1,2)],3)
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
([(2,3)],4)
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
([(3,4)],5)
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
([(2,4),(3,4)],5)
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,2,1,2,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,2,1,2,1,1] => [1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> [1,3,3,1,1] => [3,1,3,1,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,2,2,1,1] => [2,2,1,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> [2,1,2,1,1,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> [4,4,1] => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
([(0,4),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,2,1,1,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [2,1,1,2,1,1,1] => [1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,2,1,1,2,1,1] => [2,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,2),(0,3),(1,4),(1,5),(1,6),(1,7),(2,6),(2,7),(2,8),(3,4),(3,5),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> [1,1,2,1,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,7),(4,8),(5,7),(6,8),(7,8)],9)
=> [3,2,2,1,1] => [2,1,3,2,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,2,1,1,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,2,1,1,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,2,1,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,2,2,1,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,3,1,1,1,1] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,2,1,1,1,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,1,1,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,2,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,4,1,1,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,3,1,1,1] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,3,1,1,2] => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,3,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
([(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),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,2,1,1,1] => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,2,1,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,1,1,3] => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,2,1,2,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,1,2,2] => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,2,1,3,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 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),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,1,1] => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,3] => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,1,1,2,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,2,2] => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,3,1] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 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),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
([(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),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,3,2,1,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
([(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)
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
([(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),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,2,2,1,1] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
([(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)
=> [1,2,2,1,2] => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
([(0,7),(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)
=> [1,2,2,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
([(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),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,1,2,1,1] => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
([(1,6),(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)
=> [2,1,2,1,2] => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
([(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),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,3,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
([(1,5),(1,6),(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)
=> [3,3,2] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
([(0,7),(1,5),(1,6),(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)
=> [1,2,3,1,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,3,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
Description
The height of a Dyck path.
The height of a Dyck path D of semilength n is defined as the maximal height of a peak of D. The height of D at position i is the number of up-steps minus the number of down-steps before position i.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000306The bounce count of a Dyck path. St000439The position of the first down step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000662The staircase size of the code of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000141The maximum drop size of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000025The number of initial rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a Dyck path as follows:
St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000308The height of the tree associated to a permutation. St001330The hat guessing number of a graph. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001652The length of a longest interval of consecutive numbers. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000094The depth of an ordered tree. St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001589The nesting number of a perfect matching.
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