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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000845
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(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 3
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000442
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000013
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 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: St001498
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> ? = 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,4,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [4,2,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [6,2] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,2] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001933
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
St001933: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,2,2]
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,6,2,2,2,2]
=> ? = 4
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4]
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ? = 0
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [2,1] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ([(0,6),(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6),(6,2)],7)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,4,3,1,6,5,7] => ([(0,5),(0,6),(1,2),(1,3),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [2,1,5,4,6,3,7] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(3,2),(4,3),(5,3),(6,2)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [2,5,4,3,6,1,7] => ([(0,5),(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [3,5,4,2,6,1,7] => ([(0,6),(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(4,3),(5,3),(6,3)],7)
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,5,1] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [2,4,3,6,5,1,7] => ([(0,6),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [3,4,2,6,5,1,7] => ([(0,6),(1,4),(1,5),(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,5,1] => [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,7,1] => [2,3,5,6,4,1,7] => ([(0,6),(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => [2,5,4,6,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [2,1,6,5,4,3,7] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,7,1] => [3,2,6,5,4,1,7] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,1,2] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,1,2] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [2,4,3,1,5,7,6] => ([(0,6),(1,2),(1,3),(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,1,2] => [2,3,1,5,4,7,6] => ([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,1,2] => [1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,1,2] => [1,3,4,5,2,7,6] => ([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,1,2] => [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,1,2] => [2,4,3,5,1,7,6] => ([(0,2),(0,3),(1,4),(1,5),(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,1,2] => [3,4,2,5,1,7,6] => ([(0,6),(1,4),(1,5),(2,3),(3,6),(6,4),(6,5)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,1,3] => [1,3,2,4,6,7,5] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [1,4,3,5,6,7,2] => ([(0,2),(0,3),(0,4),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => [2,4,3,5,6,7,1] => ([(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => [2,1,5,4,6,7,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(3,2),(5,3),(6,3)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,1,6] => [1,3,4,6,5,7,2] => ([(0,3),(0,4),(1,6),(2,6),(4,5),(5,1),(5,2)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [3,4,2,6,5,7,1] => ([(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,1,5] => [2,1,5,6,4,7,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(5,2),(6,3)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [2,3,5,6,4,7,1] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,1,7] => [2,3,6,5,4,7,1] => ([(1,5),(2,6),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [3,2,6,5,4,7,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,1,7] => [2,4,6,5,3,7,1] => ([(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [3,4,6,5,2,7,1] => ([(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [3,5,6,4,2,7,1] => ([(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,1,2,3] => [2,3,1,4,7,6,5] => ([(0,6),(1,2),(2,6),(6,3),(6,4),(6,5)],7)
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,1,2,3] => [1,3,4,2,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1)],7)
=> ? = 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,1,2,3] => [2,3,4,1,7,6,5] => ([(0,2),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,1,2,4] => [2,3,1,5,7,6,4] => ([(0,5),(0,6),(1,4),(4,5),(4,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
=> ? = 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 86%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> []
=> => => ? = 0
[1,0,1,0]
=> [1]
=> 10 => 01 => 1
[1,1,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 011 => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 01011 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 00101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01001 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 011 => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0101011 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 01011001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0101101 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 010111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 01100101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0110011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 01101001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0110101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 011011 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 01110001 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0111001 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 01111 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0010101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 001101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00111 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 010011 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0101001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 01011 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0110001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 000101 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00011 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 00101 => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 01010101011 => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => 01010101101 => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> 1110101010 => 0101010111 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> 101001101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => 01010110101 => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> 1101101010 => 0101011011 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> 100011101010 => ? => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> 10011101010 => 01010111001 => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> 1011101010 => 0101011101 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => 01011010101 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> 1101011010 => 0101101011 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> 1011011010 => 0101101101 => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> 11000111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> 1100111010 => 0101110011 => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> 1010111010 => 0101110101 => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> 10001111010 => 01011110001 => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> 1001111010 => 0101111001 => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> 1110100110 => 0110010111 => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => 01100110011 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> 1101100110 => 0110011011 => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> 10011100110 => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> 1011100110 => 0110011101 => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> 11010010110 => 01101001011 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => 01101010101 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> 1101010110 => 0110101011 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> 1011010110 => 0110101101 => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1,1]
=> 11000110110 => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> 1100110110 => 0110110011 => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> 1010110110 => 0110110101 => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,1,1]
=> 10001110110 => ? => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> 11010001110 => 01110001011 => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1,1,1]
=> 10110001110 => ? => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> 1110001110 => 0111000111 => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,5,3,1,1,1]
=> 11001001110 => 01110010011 => ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> 1101001110 => 0111001011 => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> 10011001110 => ? => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> 1011001110 => 0111001101 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> 10100101110 => 01110100101 => ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> 10001101110 => 01110110001 => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,2,1,1,1]
=> 1001101110 => 0111011001 => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> 11000011110 => 01111000011 => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1,1,1]
=> 10100011110 => 01111000101 => ? = 4
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 86%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> []
=> => ? => ? = 0
[1,0,1,0]
=> [1]
=> 10 => 01 => 1
[1,1,0,0]
=> []
=> => ? => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 101 => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,0,0,0]
=> []
=> => ? => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 10101 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001101 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1101 => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01001 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1001 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00101 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 101 => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,1,0,0,0,0]
=> []
=> => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 1010101 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00110101 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0110101 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 110101 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 01001101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1001101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00101101 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0101101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 101101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00011101 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011101 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 11101 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0101001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 101001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 011001 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 11001 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 100101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0010101 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 10101 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0001101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001101 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 1101 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 010001 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 10001 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01001 => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => ? => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => 01101010101 => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> 1110101010 => 1101010101 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => 01011010101 => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> 1101101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> 10011101010 => ? => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> 1011101010 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => 01010110101 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> 1101011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> 1011011010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> 11000111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> 1100111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> 1010111010 => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> 10001111010 => ? => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> 1001111010 => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> 1110100110 => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => ? => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> 1101100110 => ? => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> 10011100110 => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> 1011100110 => ? => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> 11010010110 => ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => 01010101101 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> 1101010110 => ? => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> 1011010110 => ? => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1,1]
=> 11000110110 => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> 1100110110 => ? => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> 1010110110 => ? => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,1,1]
=> 10001110110 => ? => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> 11010001110 => 10100011101 => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1,1,1]
=> 10110001110 => ? => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> 1110001110 => ? => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,5,3,1,1,1]
=> 11001001110 => ? => ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> 1101001110 => 1010011101 => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> 10011001110 => ? => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> 1011001110 => ? => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> 10100101110 => ? => ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> 10001101110 => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> 11000011110 => 10000111101 => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1,1,1]
=> 10100011110 => ? => ? = 4
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1,1,1]
=> 1100011110 => 1000111101 => ? = 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1,1,1]
=> 10010011110 => ? => ? = 4
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? = 4
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000651
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => ? = 4
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [6,5,3,2,1,4,7] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,5,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,6,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [6,5,3,2,1,4,7] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,5,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 3
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St000308
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,4,3,5,2] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,4,5,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,5,3,2,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,3,4,2,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,4,2,3,5] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,2,4,3,5] => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [2,4,3,5,1] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,5,1,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [5,1,4,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [5,1,3,2,4] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => [1,6,7,5,4,3,2] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => [1,6,5,7,4,3,2] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [1,5,6,7,4,3,2] => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => [1,7,6,4,5,3,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => [1,6,5,4,7,3,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => [1,5,6,4,7,3,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [1,7,4,5,6,3,2] => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,7,5,4,6,1] => [1,6,4,5,7,3,2] => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => [1,4,6,5,7,3,2] => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => [1,4,5,6,7,3,2] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => [1,6,5,4,3,7,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,7,3,4,6,5,1] => [1,5,6,4,3,7,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => [1,4,6,5,3,7,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => [1,4,5,6,3,7,2] => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => [1,6,7,3,4,5,2] => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,7,4,3,6,5,1] => [1,5,6,3,4,7,2] => ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => [1,3,6,5,4,7,2] => ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => [1,7,3,4,5,6,2] => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => [1,6,3,4,5,7,2] => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => [1,3,6,4,5,7,2] => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => [1,3,5,4,6,7,2] => ? = 4 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => [1,7,5,6,4,2,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => [1,5,6,7,4,2,3] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => [1,7,6,4,5,2,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => [1,6,7,4,5,2,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,7,4,6,5,1] => [1,5,6,4,7,2,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => [1,7,4,5,6,2,3] => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,7,5,4,6,1] => [1,6,4,5,7,2,3] => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,7,5,6,4,1] => [1,4,6,5,7,2,3] => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,2,3,5,7,6,1] => [1,6,7,5,3,2,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => [1,5,6,4,3,2,7] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => [1,4,6,5,3,2,7] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => [1,4,5,6,3,2,7] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,2,4,3,7,6,1] => [1,6,7,3,4,2,5] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [6,2,4,3,5,7,1] => [1,7,5,3,4,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [7,2,4,3,6,5,1] => [1,5,6,3,4,2,7] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => [1,3,6,5,4,2,7] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => [1,3,5,6,4,2,7] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => [1,7,3,4,5,2,6] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [7,2,5,4,3,6,1] => [1,6,3,4,5,2,7] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [7,2,5,4,6,3,1] => [1,3,6,4,5,2,7] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => [1,3,5,4,6,2,7] => ? = 3 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => [1,3,4,5,6,2,7] => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => [1,6,7,5,2,3,4] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => [1,7,5,6,2,3,4] => ? = 3 + 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001330The hat guessing number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000983The length of the longest alternating subword. St001589The nesting number of a perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
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