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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St001120
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The length of a longest path in a graph.
Matching statistic: St000147
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ 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 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 4 = 3 + 1
Description
The largest part of an integer partition.
Matching statistic: St000381
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [2] => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [4] => 4 = 3 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4 = 3 + 1
Description
The largest part of an integer composition.
Matching statistic: St000010
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ 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%
Mp00037: Graphs —to partition of connected components⟶ 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 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [2]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
Description
The length of the partition.
Matching statistic: St000013
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
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 $i$.
Matching statistic: St000676
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ 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%
Mp00037: Graphs —to partition of connected components⟶ 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 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 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)
Mp00114: Permutations —connectivity set⟶ 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%
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] => => => ? = 0
[1,2] => 1 => 0 => 0
[2,1] => 0 => 1 => 1
[1,2,3] => 11 => 00 => 0
[1,3,2] => 10 => 01 => 1
[2,1,3] => 01 => 10 => 1
[2,3,1] => 00 => 11 => 2
[3,1,2] => 00 => 11 => 2
[3,2,1] => 00 => 11 => 2
[1,2,3,4] => 111 => 000 => 0
[1,2,4,3] => 110 => 001 => 1
[1,3,2,4] => 101 => 010 => 1
[1,3,4,2] => 100 => 011 => 2
[1,4,2,3] => 100 => 011 => 2
[1,4,3,2] => 100 => 011 => 2
[2,1,3,4] => 011 => 100 => 1
[2,1,4,3] => 010 => 101 => 1
[2,3,1,4] => 001 => 110 => 2
[2,3,4,1] => 000 => 111 => 3
[2,4,1,3] => 000 => 111 => 3
[2,4,3,1] => 000 => 111 => 3
[3,1,2,4] => 001 => 110 => 2
[3,1,4,2] => 000 => 111 => 3
[3,2,1,4] => 001 => 110 => 2
[3,2,4,1] => 000 => 111 => 3
[3,4,1,2] => 000 => 111 => 3
[3,4,2,1] => 000 => 111 => 3
[4,1,2,3] => 000 => 111 => 3
[4,1,3,2] => 000 => 111 => 3
[4,2,1,3] => 000 => 111 => 3
[4,2,3,1] => 000 => 111 => 3
[4,3,1,2] => 000 => 111 => 3
[4,3,2,1] => 000 => 111 => 3
[1,2,3,4,5] => 1111 => 0000 => 0
[1,2,3,5,4] => 1110 => 0001 => 1
[1,2,4,3,5] => 1101 => 0010 => 1
[1,2,4,5,3] => 1100 => 0011 => 2
[1,2,5,3,4] => 1100 => 0011 => 2
[1,2,5,4,3] => 1100 => 0011 => 2
[1,3,2,4,5] => 1011 => 0100 => 1
[1,3,2,5,4] => 1010 => 0101 => 1
[1,3,4,2,5] => 1001 => 0110 => 2
[1,3,4,5,2] => 1000 => 0111 => 3
[1,3,5,2,4] => 1000 => 0111 => 3
[1,3,5,4,2] => 1000 => 0111 => 3
[1,4,2,3,5] => 1001 => 0110 => 2
[1,4,2,5,3] => 1000 => 0111 => 3
[1,4,3,2,5] => 1001 => 0110 => 2
[1,4,3,5,2] => 1000 => 0111 => 3
[1,4,5,2,3] => 1000 => 0111 => 3
[1,4,5,3,2] => 1000 => 0111 => 3
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000503
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> {{1}}
=> ? = 0
[1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0
[2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
Description
The maximal difference between two elements in a common block.
Matching statistic: St000442
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> ? = 0
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 0
[2,1] => 0 => [2] => [1,1,0,0]
=> 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,3,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
Description
The maximal area to the right of an up step of a Dyck path.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001268The size of the largest ordinal summand in the poset. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000171The degree of the graph. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St000209Maximum difference of elements in cycles. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001090The number of pop-stack-sorts needed to sort 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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