Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000845
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([(0,1)],2)
 => 1
[2,1] => [2,1] => ([],2)
 => 0
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => [2,3,1] => ([(1,2)],3)
 => 1
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => 1
[3,2,1] => [3,2,1] => ([],3)
 => 0
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 1
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 1
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 2
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 2
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => 1
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => 1
[4,3,2,1] => [4,3,2,1] => ([],4)
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 0
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1] => [2,1] => ([],2)
 => ([],2)
 => 0
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 1
[2,3,1] => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[3,2,1] => [3,2,1] => ([],3)
 => ([],3)
 => 0
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 1
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => 1
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => 1
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[4,3,2,1] => [4,3,2,1] => ([],4)
 => ([],4)
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,1,6,7,5] => [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,1,7,5,6] => [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,5,1,7,6] => [4,3,2,7,1,6,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,5,7,1,6] => [4,3,2,7,6,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,6,1,5,7] => [4,3,2,7,1,6,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,6,1,7,5] => [4,3,2,7,1,6,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,6,5,1,7] => [4,3,2,7,6,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,6,7,1,5] => [4,3,2,7,6,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,7,1,5,6] => [4,3,2,7,1,6,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,7,1,6,5] => [4,3,2,7,1,6,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,7,5,1,6] => [4,3,2,7,6,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,2,7,6,1,5] => [4,3,2,7,6,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,5,2,1,7,6] => [4,3,7,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,5,2,7,1,6] => [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,6,2,1,5,7] => [4,3,7,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,6,2,1,7,5] => [4,3,7,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,6,2,5,1,7] => [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,6,2,7,1,5] => [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,7,2,1,5,6] => [4,3,7,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,7,2,1,6,5] => [4,3,7,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,7,2,5,1,6] => [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,3,7,2,6,1,5] => [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,5,3,2,1,6,7] => [4,7,3,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,5,3,2,1,7,6] => [4,7,3,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,5,3,2,6,1,7] => [4,7,3,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,5,3,2,7,1,6] => [4,7,3,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,5,3,6,2,1,7] => [4,7,3,6,2,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,5,3,7,2,1,6] => [4,7,3,6,2,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,6,3,2,1,5,7] => [4,7,3,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,6,3,2,1,7,5] => [4,7,3,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,6,3,2,5,1,7] => [4,7,3,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,6,3,2,7,1,5] => [4,7,3,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,6,3,5,2,1,7] => [4,7,3,6,2,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,6,3,7,2,1,5] => [4,7,3,6,2,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,7,3,2,1,5,6] => [4,7,3,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,7,3,2,1,6,5] => [4,7,3,2,1,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,7,3,2,5,1,6] => [4,7,3,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,7,3,2,6,1,5] => [4,7,3,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,7,3,5,2,1,6] => [4,7,3,6,2,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[4,7,3,6,2,1,5] => [4,7,3,6,2,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[5,3,2,1,4,6,7] => [5,3,2,1,7,6,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
[5,3,2,1,4,7,6] => [5,3,2,1,7,6,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 3
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[2,1] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[8,7,4,3,5,6,2,1] => [1,2,5,6,4,3,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ?
 => ? = 2 + 1
[4,3,5,6,7,8,2,1] => [5,6,4,3,2,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[7,6,8,5,3,2,4,1] => [2,3,1,4,6,7,5,8] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[8,7,6,3,2,4,5,1] => [1,2,3,6,7,5,4,8] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[8,6,4,3,5,2,7,1] => [1,3,5,6,4,7,2,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
 => ? = 2 + 1
[8,4,5,2,3,6,7,1] => [1,5,4,7,6,3,2,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[6,7,5,8,2,3,1,4] => [3,2,4,1,7,6,8,5] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[7,5,6,8,3,1,2,4] => [2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[8,7,5,3,2,4,1,6] => [1,2,4,6,7,5,8,3] => ?
 => ? = 2 + 1
[8,7,4,3,2,1,5,6] => [1,2,5,6,7,8,4,3] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[8,7,3,4,1,2,5,6] => [1,2,6,5,8,7,4,3] => [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 3 + 1
[7,4,5,2,3,6,1,8] => [2,5,4,7,6,3,8,1] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[4,3,8,7,6,5,2,1] => [5,6,1,2,3,4,7,8] => ?
 => ? = 4 + 1
[4,3,7,8,6,5,2,1] => [5,6,2,1,3,4,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[4,3,6,7,8,5,2,1] => [5,6,3,2,1,4,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[4,3,5,7,8,6,2,1] => [5,6,4,2,1,3,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[4,3,6,5,8,7,2,1] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[4,3,5,6,8,7,2,1] => [5,6,4,3,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[4,3,7,6,5,8,2,1] => [5,6,2,3,4,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[4,3,5,7,6,8,2,1] => [5,6,4,2,3,1,7,8] => ?
 => ? = 4 + 1
[4,3,6,5,7,8,2,1] => [5,6,3,4,2,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[2,1,4,7,6,8,5,3] => [7,8,5,2,3,1,4,6] => ?
 => ? = 6 + 1
[2,1,5,4,7,8,6,3] => [7,8,4,5,2,1,3,6] => ?
 => ? = 6 + 1
[2,1,5,6,4,8,7,3] => [7,8,4,3,5,1,2,6] => ?
 => ? = 6 + 1
[2,1,5,4,6,8,7,3] => [7,8,4,5,3,1,2,6] => ?
 => ? = 6 + 1
[2,1,6,7,5,4,8,3] => [7,8,3,2,4,5,1,6] => ?
 => ? = 6 + 1
[2,1,5,7,6,4,8,3] => [7,8,4,2,3,5,1,6] => ?
 => ? = 6 + 1
[2,1,4,6,7,5,8,3] => [7,8,5,3,2,4,1,6] => ?
 => ? = 6 + 1
[2,1,5,6,4,7,8,3] => [7,8,4,3,5,2,1,6] => ?
 => ? = 6 + 1
[2,1,3,5,7,6,8,4] => [7,8,6,4,2,3,1,5] => ?
 => ? = 6 + 1
[2,1,3,5,4,8,7,6] => [7,8,6,4,5,1,2,3] => ?
 => ? = 6 + 1
[2,1,3,5,4,7,8,6] => [7,8,6,4,5,2,1,3] => ?
 => ? = 6 + 1
[2,1,3,5,6,4,8,7] => [7,8,6,4,3,5,1,2] => ?
 => ? = 6 + 1
[2,1,4,5,3,6,8,7] => [7,8,5,4,6,3,1,2] => ?
 => ? = 6 + 1
[2,1,4,6,7,5,3,8] => [7,8,5,3,2,4,6,1] => ?
 => ? = 6 + 1
[2,1,5,4,6,7,3,8] => [7,8,4,5,3,2,6,1] => ?
 => ? = 6 + 1
[2,1,4,3,6,7,5,8] => [7,8,5,6,3,2,4,1] => ?
 => ? = 6 + 1
[2,1,5,6,4,3,7,8] => [7,8,4,3,5,6,2,1] => ?
 => ? = 6 + 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: St000442
(load all 4 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => ? = 0
[1,2] => [2,1] => [1,1,0,0]
 => [1,1,0,0]
 => 1
[2,1] => [1,2] => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[6,7,8,3,4,5,2,1] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ?
 => ?
 => ? = 2
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 2
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
 => ?
 => ? = 2
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 1
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[8,5,6,7,4,1,2,3] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
 => ? = 2
[8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 2
[6,7,5,8,2,3,1,4] => [3,2,4,1,7,6,8,5] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 2
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => ? = 2
[7,5,6,8,3,1,2,4] => [2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 2
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,7,5,3,2,4,1,6] => [1,2,4,6,7,5,8,3] => ?
 => ?
 => ? = 2
[5,6,4,3,1,2,7,8] => [4,3,5,6,8,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[4,5,6,1,2,3,7,8] => [5,4,3,8,7,6,2,1] => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[4,3,8,7,6,5,2,1] => [5,6,1,2,3,4,7,8] => ?
 => ?
 => ? = 4
[4,3,5,7,6,8,2,1] => [5,6,4,2,3,1,7,8] => ?
 => ?
 => ? = 4
[2,1,4,7,6,8,5,3] => [7,8,5,2,3,1,4,6] => ?
 => ?
 => ? = 6
[2,1,5,4,7,8,6,3] => [7,8,4,5,2,1,3,6] => ?
 => ?
 => ? = 6
[2,1,5,6,4,8,7,3] => [7,8,4,3,5,1,2,6] => ?
 => ?
 => ? = 6
[2,1,5,4,6,8,7,3] => [7,8,4,5,3,1,2,6] => ?
 => ?
 => ? = 6
[2,1,6,7,5,4,8,3] => [7,8,3,2,4,5,1,6] => ?
 => ?
 => ? = 6
[2,1,5,7,6,4,8,3] => [7,8,4,2,3,5,1,6] => ?
 => ?
 => ? = 6
[2,1,4,6,7,5,8,3] => [7,8,5,3,2,4,1,6] => ?
 => ?
 => ? = 6
[2,1,5,6,4,7,8,3] => [7,8,4,3,5,2,1,6] => ?
 => ?
 => ? = 6
[2,1,3,5,7,6,8,4] => [7,8,6,4,2,3,1,5] => ?
 => ?
 => ? = 6
[2,1,3,5,4,8,7,6] => [7,8,6,4,5,1,2,3] => ?
 => ?
 => ? = 6
[2,1,3,5,4,7,8,6] => [7,8,6,4,5,2,1,3] => ?
 => ?
 => ? = 6
[2,1,3,5,6,4,8,7] => [7,8,6,4,3,5,1,2] => ?
 => ?
 => ? = 6
[2,1,4,5,3,6,8,7] => [7,8,5,4,6,3,1,2] => ?
 => ?
 => ? = 6
[2,1,4,6,7,5,3,8] => [7,8,5,3,2,4,6,1] => ?
 => ?
 => ? = 6
[2,1,5,4,6,7,3,8] => [7,8,4,5,3,2,6,1] => ?
 => ?
 => ? = 6
[2,1,4,3,6,7,5,8] => [7,8,5,6,3,2,4,1] => ?
 => ?
 => ? = 6
[2,1,5,6,4,3,7,8] => [7,8,4,3,5,6,2,1] => ?
 => ?
 => ? = 6
[2,1,5,4,3,6,7,8] => [7,8,4,5,6,3,2,1] => ?
 => ?
 => ? = 6
[2,1,4,5,3,6,7,8] => [7,8,5,4,6,3,2,1] => ?
 => ?
 => ? = 6
[2,1,3,4,8,6,7,5] => [7,8,6,5,1,3,2,4] => ?
 => ?
 => ? = 6
[2,1,3,8,6,5,7,4] => [7,8,6,1,3,4,2,5] => ?
 => ?
 => ? = 6
[2,1,4,8,5,6,7,3] => [7,8,5,1,4,3,2,6] => ?
 => ?
 => ? = 6
[2,1,4,8,5,7,6,3] => [7,8,5,1,4,2,3,6] => ?
 => ?
 => ? = 6
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000451
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => 2 = 1 + 1
[2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,3,2] => [3,1,2] => 3 = 2 + 1
[1,3,2] => [1,3,2] => [3,1,2] => 3 = 2 + 1
[2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,3,1] => [2,3,1] => [2,1,3] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3 = 2 + 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 3 = 2 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 3 = 2 + 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 3 = 2 + 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 1 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 2 = 1 + 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 3 = 2 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => 5 = 4 + 1
[3,4,5,2,1,6,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,4,5,2,1,7,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,4,6,2,1,5,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,4,6,2,1,7,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,4,7,2,1,5,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,4,7,2,1,6,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,5,4,2,1,6,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,5,4,2,1,7,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,5,6,2,1,4,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,5,6,2,1,7,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,5,7,2,1,4,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,5,7,2,1,6,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,6,4,2,1,5,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,6,4,2,1,7,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,6,5,2,1,4,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,6,5,2,1,7,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,6,7,2,1,4,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,6,7,2,1,5,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,7,4,2,1,5,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,7,4,2,1,6,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,7,5,2,1,4,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,7,5,2,1,6,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,7,6,2,1,4,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[3,7,6,2,1,5,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4 + 1
[4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3 + 1
[4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3 + 1
[4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3 + 1
[4,3,2,1,6,7,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3 + 1
[4,3,2,1,7,5,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3 + 1
[4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3 + 1
[4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3 + 1
[4,3,2,5,1,7,6] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3 + 1
[4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3 + 1
[4,3,2,5,7,1,6] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3 + 1
[4,3,2,6,1,5,7] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3 + 1
[4,3,2,6,1,7,5] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3 + 1
[4,3,2,6,5,1,7] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3 + 1
[4,3,2,6,7,1,5] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3 + 1
[4,3,2,7,1,5,6] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3 + 1
[4,3,2,7,1,6,5] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3 + 1
[4,3,2,7,5,1,6] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3 + 1
[4,3,2,7,6,1,5] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3 + 1
[4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => [4,5,1,6,7,2,3] => ? = 3 + 1
[4,3,5,2,1,7,6] => [4,3,7,2,1,6,5] => [4,5,1,6,7,2,3] => ? = 3 + 1
[4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => [4,5,1,6,2,7,3] => ? = 3 + 1
[4,3,5,2,7,1,6] => [4,3,7,2,6,1,5] => [4,5,1,6,2,7,3] => ? = 3 + 1
[4,3,5,6,2,1,7] => [4,3,7,6,2,1,5] => [4,5,1,2,6,7,3] => ? = 3 + 1
[4,3,5,6,2,7,1] => [4,3,7,6,2,5,1] => [4,5,1,2,6,3,7] => ? = 3 + 1
[4,3,5,7,2,1,6] => [4,3,7,6,2,1,5] => [4,5,1,2,6,7,3] => ? = 3 + 1
[4,3,5,7,2,6,1] => [4,3,7,6,2,5,1] => [4,5,1,2,6,3,7] => ? = 3 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001039
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => ? = 0 + 1
[1,2] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[6,5,7,8,4,3,2,1] => [3,4,2,1,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[8,5,4,6,7,3,2,1] => [1,4,5,3,2,6,7,8] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 1
[7,6,8,4,3,5,2,1] => [2,3,1,5,6,4,7,8] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 1 + 1
[8,7,4,3,5,6,2,1] => [1,2,5,6,4,3,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 1
[7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ?
 => ?
 => ? = 2 + 1
[6,5,4,3,7,8,2,1] => [3,4,5,6,2,1,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[4,3,5,6,7,8,2,1] => [5,6,4,3,2,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 1 + 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[7,6,8,5,3,2,4,1] => [2,3,1,4,6,7,5,8] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 1
[8,7,6,3,2,4,5,1] => [1,2,3,6,7,5,4,8] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[8,6,4,3,5,2,7,1] => [1,3,5,6,4,7,2,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
 => ?
 => ? = 2 + 1
[8,3,2,4,5,6,7,1] => [1,6,7,5,4,3,2,8] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 4 + 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[6,5,7,8,3,4,1,2] => [3,4,2,1,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 1
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 1
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,5,6,8,3,1,2,4] => [2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 2 + 1
[8,7,6,5,2,1,3,4] => [1,2,3,4,7,8,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[8,7,5,3,2,4,1,6] => [1,2,4,6,7,5,8,3] => ?
 => ?
 => ? = 2 + 1
[8,7,4,3,2,1,5,6] => [1,2,5,6,7,8,4,3] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[8,7,3,4,1,2,5,6] => [1,2,6,5,8,7,4,3] => [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 3 + 1
[8,7,2,1,3,4,5,6] => [1,2,7,8,6,5,4,3] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 4 + 1
[7,6,4,3,5,2,1,8] => [2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 2 + 1
[7,5,4,3,2,6,1,8] => [2,4,5,6,7,3,8,1] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 1
[7,4,5,2,3,6,1,8] => [2,5,4,7,6,3,8,1] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 3 + 1
[6,5,4,3,2,1,7,8] => [3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 2 + 1
[5,4,6,2,1,3,7,8] => [4,5,3,7,8,6,2,1] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 3 + 1
[5,3,2,4,1,6,7,8] => [4,6,7,5,8,3,2,1] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,2,1,5,6,7,8] => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 4 + 1
[2,1,3,4,5,6,7,8] => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[6,5,8,7,4,3,2,1] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[4,3,8,7,6,5,2,1] => [5,6,1,2,3,4,7,8] => ?
 => ?
 => ? = 4 + 1
[4,3,7,8,6,5,2,1] => [5,6,2,1,3,4,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[4,3,6,7,8,5,2,1] => [5,6,3,2,1,4,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[4,3,5,7,8,6,2,1] => [5,6,4,2,1,3,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[6,5,4,3,8,7,2,1] => [3,4,5,6,1,2,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[4,3,6,5,8,7,2,1] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[4,3,5,6,8,7,2,1] => [5,6,4,3,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[4,3,7,6,5,8,2,1] => [5,6,2,3,4,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[4,3,5,7,6,8,2,1] => [5,6,4,2,3,1,7,8] => ?
 => ?
 => ? = 4 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000141
(load all 10 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,1,2] => 2
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 3
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 3
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 3
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 3
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[3,4,5,2,1,6,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,4,5,2,1,7,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,4,6,2,1,5,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,4,6,2,1,7,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,4,7,2,1,5,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,4,7,2,1,6,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,5,4,2,1,6,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,5,4,2,1,7,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,5,6,2,1,4,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,5,6,2,1,7,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,5,7,2,1,4,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,5,7,2,1,6,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,6,4,2,1,5,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,6,4,2,1,7,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,6,5,2,1,4,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,6,5,2,1,7,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,6,7,2,1,4,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,6,7,2,1,5,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,7,4,2,1,5,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,7,4,2,1,6,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,7,5,2,1,4,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,7,5,2,1,6,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,7,6,2,1,4,5] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[3,7,6,2,1,5,4] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 4
[4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3
[4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3
[4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3
[4,3,2,1,6,7,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3
[4,3,2,1,7,5,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3
[4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 3
[4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3
[4,3,2,5,1,7,6] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3
[4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3
[4,3,2,5,7,1,6] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3
[4,3,2,6,1,5,7] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3
[4,3,2,6,1,7,5] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3
[4,3,2,6,5,1,7] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3
[4,3,2,6,7,1,5] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3
[4,3,2,7,1,5,6] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3
[4,3,2,7,1,6,5] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 3
[4,3,2,7,5,1,6] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3
[4,3,2,7,6,1,5] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 3
[4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => [4,5,1,6,7,2,3] => ? = 3
[4,3,5,2,1,7,6] => [4,3,7,2,1,6,5] => [4,5,1,6,7,2,3] => ? = 3
[4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => [4,5,1,6,2,7,3] => ? = 3
[4,3,5,2,7,1,6] => [4,3,7,2,6,1,5] => [4,5,1,6,2,7,3] => ? = 3
[4,3,5,6,2,1,7] => [4,3,7,6,2,1,5] => [4,5,1,2,6,7,3] => ? = 3
[4,3,5,6,2,7,1] => [4,3,7,6,2,5,1] => [4,5,1,2,6,3,7] => ? = 3
[4,3,5,7,2,1,6] => [4,3,7,6,2,1,5] => [4,5,1,2,6,7,3] => ? = 3
[4,3,5,7,2,6,1] => [4,3,7,6,2,5,1] => [4,5,1,2,6,3,7] => ? = 3
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000028
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 2
[1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 3
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 3
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 3
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 3
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 3
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 2
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 4
[2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,5,7,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,6,5,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,6,7,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,4,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,4,7,6,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,4,6,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,5,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,4,7,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,6,4,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,6,7,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,5,7,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,7,4,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,5,7,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,5,7,6,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,6,4,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,6,4,7,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,6,5,4,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,6,5,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,6,5,7,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,6,7,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,6,7,4,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,6,7,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,6,7,5,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,7,4,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,7,4,5,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,7,4,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,7,4,6,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,7,5,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,7,5,4,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,7,5,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,7,5,6,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,7,6,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,7,6,4,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,3,7,6,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,3,7,6,5,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
[2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 5
[2,4,3,5,6,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 5
Description
The number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St001203
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,2] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0 + 1
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[6,5,7,8,4,3,2,1] => [3,4,2,1,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,5,4,6,7,3,2,1] => [1,4,5,3,2,6,7,8] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[7,6,8,4,3,5,2,1] => [2,3,1,5,6,4,7,8] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => ? = 1 + 1
[6,7,8,3,4,5,2,1] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
[8,7,4,3,5,6,2,1] => [1,2,5,6,4,3,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ?
 => ?
 => ? = 2 + 1
[6,5,4,3,7,8,2,1] => [3,4,5,6,2,1,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 1
[4,3,5,6,7,8,2,1] => [5,6,4,3,2,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 4 + 1
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[7,6,8,5,3,2,4,1] => [2,3,1,4,6,7,5,8] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
 => ? = 1 + 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
 => ? = 1 + 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[8,7,6,3,2,4,5,1] => [1,2,3,6,7,5,4,8] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[8,6,4,3,5,2,7,1] => [1,3,5,6,4,7,2,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 2 + 1
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
 => ?
 => ? = 2 + 1
[8,4,5,2,3,6,7,1] => [1,5,4,7,6,3,2,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 3 + 1
[8,3,2,4,5,6,7,1] => [1,6,7,5,4,3,2,8] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 4 + 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
[6,5,7,8,3,4,1,2] => [3,4,2,1,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[7,8,4,3,5,6,1,2] => [2,1,5,6,4,3,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 1 + 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2 + 1
[8,5,6,7,4,1,2,3] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 1 + 1
[6,7,5,8,2,3,1,4] => [3,2,4,1,7,6,8,5] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 2 + 1
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
 => ? = 2 + 1
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 2 + 1
[7,5,6,8,3,1,2,4] => [2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => ? = 2 + 1
[8,7,6,5,2,1,3,4] => [1,2,3,4,7,8,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[7,8,5,6,2,1,3,4] => [2,1,4,3,7,8,6,5] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[8,7,5,3,2,4,1,6] => [1,2,4,6,7,5,8,3] => ?
 => ?
 => ? = 2 + 1
[8,7,4,3,2,1,5,6] => [1,2,5,6,7,8,4,3] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 1
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St000730
(load all 3 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => ? = 0
[1,2] => [2,1] => {{1,2}}
 => 1
[2,1] => [1,2] => {{1},{2}}
 => 0
[1,2,3] => [3,2,1] => {{1,3},{2}}
 => 2
[1,3,2] => [3,1,2] => {{1,3},{2}}
 => 2
[2,1,3] => [2,3,1] => {{1,2,3}}
 => 1
[2,3,1] => [2,1,3] => {{1,2},{3}}
 => 1
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => 1
[3,2,1] => [1,2,3] => {{1},{2},{3}}
 => 0
[1,2,3,4] => [4,3,2,1] => {{1,4},{2,3}}
 => 3
[1,2,4,3] => [4,3,1,2] => {{1,4},{2,3}}
 => 3
[1,3,2,4] => [4,2,3,1] => {{1,4},{2},{3}}
 => 3
[1,3,4,2] => [4,2,1,3] => {{1,4},{2},{3}}
 => 3
[1,4,2,3] => [4,1,3,2] => {{1,4},{2},{3}}
 => 3
[1,4,3,2] => [4,1,2,3] => {{1,4},{2},{3}}
 => 3
[2,1,3,4] => [3,4,2,1] => {{1,3},{2,4}}
 => 2
[2,1,4,3] => [3,4,1,2] => {{1,3},{2,4}}
 => 2
[2,3,1,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 2
[2,3,4,1] => [3,2,1,4] => {{1,3},{2},{4}}
 => 2
[2,4,1,3] => [3,1,4,2] => {{1,3,4},{2}}
 => 2
[2,4,3,1] => [3,1,2,4] => {{1,3},{2},{4}}
 => 2
[3,1,2,4] => [2,4,3,1] => {{1,2,4},{3}}
 => 2
[3,1,4,2] => [2,4,1,3] => {{1,2,4},{3}}
 => 2
[3,2,1,4] => [2,3,4,1] => {{1,2,3,4}}
 => 1
[3,2,4,1] => [2,3,1,4] => {{1,2,3},{4}}
 => 1
[3,4,1,2] => [2,1,4,3] => {{1,2},{3,4}}
 => 1
[3,4,2,1] => [2,1,3,4] => {{1,2},{3},{4}}
 => 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 2
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 2
[4,2,1,3] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
[4,2,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[4,3,1,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 4
[1,2,3,5,4] => [5,4,3,1,2] => {{1,5},{2,4},{3}}
 => 4
[1,2,4,3,5] => [5,4,2,3,1] => {{1,5},{2,4},{3}}
 => 4
[1,2,4,5,3] => [5,4,2,1,3] => {{1,5},{2,4},{3}}
 => 4
[1,2,5,3,4] => [5,4,1,3,2] => {{1,5},{2,4},{3}}
 => 4
[1,2,5,4,3] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
 => 4
[1,3,2,4,5] => [5,3,4,2,1] => {{1,5},{2,3,4}}
 => 4
[1,3,2,5,4] => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => 4
[1,3,4,2,5] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 4
[1,3,4,5,2] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
 => 4
[1,3,5,2,4] => [5,3,1,4,2] => {{1,5},{2,3},{4}}
 => 4
[1,3,5,4,2] => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => 4
[1,4,2,3,5] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
 => 4
[1,4,2,5,3] => [5,2,4,1,3] => {{1,5},{2},{3,4}}
 => 4
[1,4,3,2,5] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 4
[1,4,3,5,2] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
 => 4
[1,4,5,2,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
 => 4
[1,4,5,3,2] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
 => 4
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 0
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => ?
 => ? = 1
[6,5,7,8,4,3,2,1] => [3,4,2,1,5,6,7,8] => ?
 => ? = 2
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => ?
 => ? = 1
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
 => ? = 1
[8,5,4,6,7,3,2,1] => [1,4,5,3,2,6,7,8] => ?
 => ? = 2
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => ?
 => ? = 1
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => ?
 => ? = 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
 => ? = 1
[7,6,8,4,3,5,2,1] => [2,3,1,5,6,4,7,8] => ?
 => ? = 1
[6,7,8,3,4,5,2,1] => [3,2,1,6,5,4,7,8] => ?
 => ? = 2
[8,7,4,3,5,6,2,1] => [1,2,5,6,4,3,7,8] => ?
 => ? = 2
[7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ?
 => ? = 2
[6,5,4,3,7,8,2,1] => [3,4,5,6,2,1,7,8] => ?
 => ? = 2
[4,3,5,6,7,8,2,1] => [5,6,4,3,2,1,7,8] => ?
 => ? = 4
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => ?
 => ? = 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
 => ? = 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => ?
 => ? = 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
 => ? = 1
[7,6,8,5,3,2,4,1] => [2,3,1,4,6,7,5,8] => ?
 => ? = 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
 => ? = 2
[8,7,6,3,2,4,5,1] => [1,2,3,6,7,5,4,8] => ?
 => ? = 2
[8,6,4,3,5,2,7,1] => [1,3,5,6,4,7,2,8] => ?
 => ? = 2
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
 => ? = 2
[8,4,5,2,3,6,7,1] => [1,5,4,7,6,3,2,8] => ?
 => ? = 3
[8,3,2,4,5,6,7,1] => [1,6,7,5,4,3,2,8] => ?
 => ? = 4
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => ?
 => ? = 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => ?
 => ? = 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => ?
 => ? = 1
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => ?
 => ? = 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 1
[6,5,7,8,3,4,1,2] => [3,4,2,1,6,5,8,7] => {{1,3},{2,4},{5,6},{7,8}}
 => ? = 2
[7,8,4,3,5,6,1,2] => [2,1,5,6,4,3,8,7] => {{1,2},{3,5},{4,6},{7,8}}
 => ? = 2
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => {{1,3},{2},{4},{5},{6,8},{7}}
 => ? = 2
[8,5,6,7,4,1,2,3] => [1,4,3,2,5,8,7,6] => ?
 => ? = 2
[8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => ?
 => ? = 2
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => {{1,2,3,4},{5,6,7,8}}
 => ? = 1
[6,7,5,8,2,3,1,4] => [3,2,4,1,7,6,8,5] => {{1,3,4},{2},{5,7,8},{6}}
 => ? = 2
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => {{1,2,4},{3},{5,7,8},{6}}
 => ? = 2
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => {{1,3,4},{2},{5,6,8},{7}}
 => ? = 2
[7,5,6,8,3,1,2,4] => [2,4,3,1,6,8,7,5] => {{1,2,4},{3},{5,6,8},{7}}
 => ? = 2
[8,7,6,5,2,1,3,4] => [1,2,3,4,7,8,6,5] => {{1},{2},{3},{4},{5,7},{6,8}}
 => ? = 2
[7,8,5,6,2,1,3,4] => [2,1,4,3,7,8,6,5] => {{1,2},{3,4},{5,7},{6,8}}
 => ? = 2
[6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => {{1,3},{2,4},{5,7},{6,8}}
 => ? = 2
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => {{1,4},{2,3},{5,8},{6,7}}
 => ? = 3
[8,7,5,3,2,4,1,6] => [1,2,4,6,7,5,8,3] => ?
 => ? = 2
Description
The maximal arc length of a set partition. The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000527The width of the poset. St000094The depth of an ordered tree. St000662The staircase size of the code of a permutation. St000306The bounce count of a Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000651The maximal size of a rise in a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001589The nesting number of a perfect matching.