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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [] => 0
[.,[.,.]]
=> [2,1] => [1] => 1
[[.,.],.]
=> [1,2] => [1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000069
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 82% ●values known / values provided: 92%●distinct values known / distinct values provided: 82%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 82% ●values known / values provided: 92%●distinct values known / distinct values provided: 82%
Values
[.,.]
=> [1] => [] => ([],0)
=> 0
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[[[.,.],.],.],[[[.,.],.],.]],.]
=> [5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 2
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [9,10,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[[.,.],[[[[[[[.,.],.],.],.],.],.],.]]
=> [3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 2
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
=> ? = 2
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> [9,8,10,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> [9,8,7,10,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> [9,8,7,6,10,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> [9,8,7,6,5,10,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> [9,8,7,6,5,4,10,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> [9,8,7,6,5,4,3,10,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? = 7
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> [9,8,7,6,5,4,3,2,10,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 9
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [9,10,7,8,5,6,3,4,1,2] => [9,7,8,5,6,3,4,1,2] => ([(1,8),(2,7),(3,6),(4,5)],9)
=> ? = 5
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => ([(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
=> ? = 2
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [4,2,3,5,6,7,8,9,1] => [4,2,3,5,6,7,8,1] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 2
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 2
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => ([(1,8),(2,8),(4,6),(5,4),(6,3),(7,5),(8,7)],9)
=> ? = 2
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => ([(1,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],10)
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> [8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> [7,8,9,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 2
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> [6,7,8,9,1,2,3,4,5] => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
=> [9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> [9,10,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> [8,9,10,1,2,3,4,5,6,7] => [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> [7,8,9,10,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
=> ? = 2
[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]]
=> [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
=> ? = 2
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [9,8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [10,9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,1,2] => ([(7,8)],9)
=> ? = 8
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([],10)
=> ? = 10
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => ([(5,7),(6,7)],8)
=> ? = 6
[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [9,8,7,6,5,4,1,2,3] => [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
=> ? = 6
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ? = 1
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
=> ? = 1
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(8,7)],9)
=> ? = 1
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,6)],9)
=> ? = 1
[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(7,6),(8,5)],9)
=> ? = 1
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(6,5),(7,6),(8,4)],9)
=> ? = 1
Description
The number of maximal elements of a poset.
Matching statistic: St000234
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 90%●distinct values known / distinct values provided: 64%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 90%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [] => [] => ? = 0 - 1
[.,[.,.]]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [2,1,5,4,7,6,3] => ? = 2 - 1
[[[[.,.],.],[[[.,[.,.]],.],.]],.]
=> [5,4,6,7,1,2,3,8] => [5,4,6,7,1,2,3] => [3,4,2,1,7,6,5] => ? = 2 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => ? = 2 - 1
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [9,10,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[[.,.],[[[[[[[.,.],.],.],.],.],.],.]]
=> [3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [6,5,4,3,2,1,8,7] => ? = 2 - 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [8,7,6,5,4,3,2,1,9] => ? = 2 - 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ? = 1 - 1
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 8 - 1
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> [9,8,10,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> [9,8,7,10,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> [9,8,7,6,10,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> [9,8,7,6,5,10,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> [9,8,7,6,5,4,10,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> [9,8,7,6,5,4,3,10,2,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => ? = 7 - 1
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> [9,8,7,6,5,4,3,2,10,1] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 9 - 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [9,10,7,8,5,6,3,4,1,2] => [9,7,8,5,6,3,4,1,2] => [1,3,2,5,4,7,6,9,8] => ? = 5 - 1
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ? = 1 - 1
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ? = 1 - 1
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> [9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [7,8,6,5,4,3,2,1] => ? = 1 - 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [6,7,5,4,3,2,1,8] => ? = 2 - 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => ? = 1 - 1
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> [10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [8,9,7,6,5,4,3,2,1] => ? = 1 - 1
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [4,2,3,5,6,7,8,9,1] => [4,2,3,5,6,7,8,1] => [5,7,6,4,3,2,1,8] => ? = 2 - 1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => [6,5,7,4,3,2,1,8] => ? = 2 - 1
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [7,8,6,5,4,3,2,1,9] => ? = 2 - 1
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => [9,8,7,6,5,4,3,2,1,10] => ? = 2 - 1
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> [8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => ? = 2 - 1
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> [7,8,9,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [2,1,8,7,6,5,4,3] => ? = 2 - 1
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> [6,7,8,9,1,2,3,4,5] => [6,7,8,1,2,3,4,5] => [3,2,1,8,7,6,5,4] => ? = 2 - 1
[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
=> [9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => ? = 2 - 1
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> [9,10,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => ? = 2 - 1
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> [8,9,10,1,2,3,4,5,6,7] => [8,9,1,2,3,4,5,6,7] => [2,1,9,8,7,6,5,4,3] => ? = 2 - 1
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> [7,8,9,10,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => [3,2,1,9,8,7,6,5,4] => ? = 2 - 1
[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]]
=> [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [4,3,2,1,9,8,7,6,5] => ? = 2 - 1
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [9,8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => ? = 7 - 1
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [10,9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,9,8] => ? = 8 - 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => ? = 10 - 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000908
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 64% ●values known / values provided: 83%●distinct values known / distinct values provided: 64%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 64% ●values known / values provided: 83%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [] => ([],0)
=> ? = 0
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 4
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[.,[[[.,.],.],.]],.],[[.,.],.]]
=> [7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [7,8,3,4,1,2,5,6] => [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 2
[[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [8,5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> [8,5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 1
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [8,2,3,4,1,5,6,7] => [2,3,4,1,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],[[.,.],[.,.]]],.]
=> [7,5,6,1,2,3,4,8] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[[.,.],.],.],[[[.,.],.],.]],.]
=> [5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 1
[[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [4,5,6,1,2,3,7,8] => [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 1
[[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000914
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 64% ●values known / values provided: 83%●distinct values known / distinct values provided: 64%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 64% ●values known / values provided: 83%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [] => ([],0)
=> ? = 0
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> ? = 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> ? = 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 4
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => ([(2,3)],4)
=> 3
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[.,[[[.,.],.],.]],.],[[.,.],.]]
=> [7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [7,8,3,4,1,2,5,6] => [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 2
[[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [8,5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> [8,5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 1
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [8,2,3,4,1,5,6,7] => [2,3,4,1,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[[[[.,.],.],.],[[.,.],[.,.]]],.]
=> [7,5,6,1,2,3,4,8] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[[[[.,.],.],.],[[[.,.],.],.]],.]
=> [5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 1
[[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 1
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St000740
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 81%●distinct values known / distinct values provided: 64%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 81%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [] => [] => ? = 0
[.,[.,.]]
=> [2,1] => [1] => [1] => 1
[[.,.],.]
=> [1,2] => [1] => [1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [3,1,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [3,2,1] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [2,3,1] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,2,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,4,2,3] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,2,3] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,4,2,3] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,3,1,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,3,4,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,4,2,1] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [4,2,3,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 4
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [4,5,6,3,7,1,2] => ? = 2
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [6,3,4,5,7,1,2] => ? = 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [5,4,3,6,7,1,2] => ? = 2
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [4,5,3,6,7,1,2] => ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [5,3,4,6,7,1,2] => ? = 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 2
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => ? = 2
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? = 4
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? = 4
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? = 4
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [7,6,1,5,2,4,3] => ? = 3
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [4,5,6,7,1,3,2] => ? = 2
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [4,5,6,7,1,3,2] => ? = 2
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [7,1,6,2,4,5,3] => ? = 3
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,6,7,2,4,5,3] => ? = 3
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,6,7,2,4,5,3] => ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [7,1,6,2,4,5,3] => ? = 3
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,7,2,4,5,6,3] => ? = 3
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,7,2,4,5,6,3] => ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [6,7,1,3,4,5,2] => ? = 2
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [1,7,2,4,5,6,3] => ? = 3
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [6,7,1,3,4,5,2] => ? = 2
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [7,1,5,6,3,4,2] => ? = 2
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [7,1,3,4,5,6,2] => ? = 2
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [7,1,3,4,5,6,2] => ? = 2
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,7,3,4,5,6,2] => ? = 2
[[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,7,3,4,5,6,2] => ? = 2
[[[.,[[.,[.,.]],.]],.],[[.,.],.]]
=> [7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [1,5,4,6,3,7,2] => ? = 2
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [7,8,3,4,1,2,5,6] => [7,3,4,1,2,5,6] => [1,5,6,3,4,7,2] => ? = 2
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [7,8,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [1,6,4,3,5,7,2] => ? = 2
[[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => [1,6,3,4,5,7,2] => ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [1,5,4,3,6,7,2] => ? = 2
[[[[.,[.,[.,.]]],.],.],[[.,.],.]]
=> [7,8,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [1,5,4,3,6,7,2] => ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [7,6,5,4,3,2,1] => ? = 1
[[.,[.,[.,[[[.,.],.],.]]]],[.,.]]
=> [8,4,5,6,3,2,1,7] => [4,5,6,3,2,1,7] => [5,6,7,4,3,2,1] => ? = 1
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [8,5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => [6,7,4,5,2,3,1] => ? = 1
[[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> [8,5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => [6,7,2,3,4,5,1] => ? = 1
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [5,4,3,2,6,7,1] => ? = 1
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [8,2,3,4,1,5,6,7] => [2,3,4,1,5,6,7] => [3,4,5,2,6,7,1] => ? = 1
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [5,2,3,4,6,7,1] => ? = 1
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [4,3,2,5,6,7,1] => ? = 1
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => [3,4,2,5,6,7,1] => ? = 1
[[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => [4,2,3,5,6,7,1] => ? = 1
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 1
[[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> [3,2,4,5,6,7,1,8] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 2
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => ? = 2
[[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3] => [1,6,7,2,4,5,3] => ? = 3
[[[[.,.],.],[[[.,[.,.]],.],.]],.]
=> [5,4,6,7,1,2,3,8] => [5,4,6,7,1,2,3] => [6,5,7,1,3,4,2] => ? = 2
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000989
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 79%●distinct values known / distinct values provided: 64%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 79%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1,0]
=> [1] => [] => ? = 0 - 1
[.,[.,.]]
=> [1,1,0,0]
=> [1,2] => [1] => ? = 1 - 1
[[.,.],.]
=> [1,0,1,0]
=> [2,1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 0 = 1 - 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3 = 4 - 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 2 = 3 - 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 2 = 3 - 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? = 7 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,5,1,6,7,8,2,3] => ? => ? = 2 - 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,6,7,8,2,5] => ? => ? = 2 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,8,2,3] => [4,1,5,6,7,2,3] => ? = 2 - 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,8,2,4] => ? => ? = 2 - 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => ? = 2 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,3,5,6,7,8] => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,6,1,3,5,7,8] => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [2,4,6,1,3,5,7] => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1,3,8,5,6,7] => [2,4,1,3,5,6,7] => ? = 5 - 1
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => [2,4,6,1,3,5,7] => ? = 4 - 1
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,4,1,7,8,3,5,6] => ? => ? = 3 - 1
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,7,1,3,8] => ? => ? = 2 - 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,1,3,4,5,8,6,7] => ? => ? = 6 - 1
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,1,3] => [2,4,5,6,7,1,3] => ? = 2 - 1
[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => ? => ? = 5 - 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,5,1,4,6,7,8] => ? => ? = 4 - 1
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6,8] => ? => ? = 3 - 1
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4,7,8] => ? => ? = 3 - 1
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,8,1,4,7] => ? => ? = 3 - 1
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,7,1,8,4,6] => ? => ? = 3 - 1
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,8,5,6,7] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7,8] => ? => ? = 3 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,8,1,5,7] => ? => ? = 3 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,7,1,5,8] => ? => ? = 2 - 1
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,4,1,5,8,6,7] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,1,8,5,7] => ? => ? = 3 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [2,3,4,6,7,1,5] => ? = 2 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,7,8,3,6] => ? => ? = 2 - 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [2,3,4,5,1,6,7] => ? = 3 - 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [2,3,4,5,1,6,7] => ? = 3 - 1
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6,8] => ? => ? = 2 - 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,5,1,8,6,7] => [2,3,4,5,1,6,7] => ? = 3 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [2,3,4,5,7,1,6] => ? = 2 - 1
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,6,5,7,8] => [2,3,4,1,6,5,7] => ? = 2 - 1
[[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,8,5,7] => [2,3,4,1,6,5,7] => ? = 2 - 1
[[[.,[[.,[.,.]],.]],.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,4,2,5,6,8,3,7] => ? => ? = 2 - 1
[[[.,[[[.,.],.],.]],.],[[.,.],.]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,5,6,8,2,7] => ? => ? = 2 - 1
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => ? => ? = 2 - 1
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,5,6,8,4,7] => ? => ? = 2 - 1
[[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,5,6,4,7,8] => ? => ? = 2 - 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,6,3,7,8] => ? => ? = 2 - 1
[[[[.,[.,[.,.]]],.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,6,8,3,7] => ? => ? = 2 - 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 78%●distinct values known / distinct values provided: 64%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 78%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [] => [] => ? = 0 - 1
[.,[.,.]]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [2,1] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [4,1,2,3] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [4,3,1,2] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,2,3] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,3,1,2] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [4,3,2,1] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,4,2,1] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [4,5,6,3,7,1,2] => ? = 2 - 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [2,3,5,4,6,7,8,1] => ? => ? => ? = 2 - 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [5,4,3,6,7,1,2] => ? = 2 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [4,5,3,6,7,1,2] => ? = 2 - 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [2,4,3,5,6,7,8,1] => ? => ? => ? = 2 - 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 2 - 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => ? = 2 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => ? => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,3,5,8,7,6,4,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,3,5,7,8,6,4,2] => [1,3,5,7,6,4,2] => [2,4,6,1,3,5,7] => ? = 4 - 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ? => ? => ? = 5 - 1
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,3,5,7,6,8,4,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [1,3,6,5,7,8,4,2] => [1,3,6,5,7,4,2] => [2,4,7,6,1,3,5] => ? = 3 - 1
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => [2,4,5,6,7,1,3] => ? = 2 - 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? => ? => ? = 6 - 1
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => [2,4,5,6,7,1,3] => ? = 2 - 1
[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,8,7,6,5,4,3] => ? => ? => ? = 5 - 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,2,4,8,7,6,5,3] => [1,2,4,7,6,5,3] => [2,3,5,1,4,6,7] => ? = 4 - 1
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,2,4,6,8,7,5,3] => ? => ? => ? = 3 - 1
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,2,4,5,8,7,6,3] => ? => ? => ? = 3 - 1
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,2,4,5,7,8,6,3] => ? => ? => ? = 3 - 1
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ? => ? => ? = 3 - 1
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,8,7,6,5,4] => [1,2,3,7,6,5,4] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,2,3,7,8,6,5,4] => [1,2,3,7,6,5,4] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,2,3,7,6,8,5,4] => [1,2,3,7,6,5,4] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => ? = 3 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => ? = 3 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,2,3,5,6,8,7,4] => [1,2,3,5,6,7,4] => [2,3,4,6,7,1,5] => ? = 2 - 1
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,2,3,7,6,5,8,4] => [1,2,3,7,6,5,4] => [2,3,4,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,2,3,5,7,6,8,4] => [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => ? = 3 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => [2,3,4,6,7,1,5] => ? = 2 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => [2,4,5,3,7,1,6] => ? = 2 - 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,5] => [2,3,4,5,1,6,7] => ? = 3 - 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,2,3,4,7,8,6,5] => [1,2,3,4,7,6,5] => [2,3,4,5,1,6,7] => ? = 3 - 1
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,2,3,4,6,8,7,5] => ? => ? => ? = 2 - 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,7,6,5] => [2,3,4,5,1,6,7] => ? = 3 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,2,3,5,4,8,7,6] => ? => ? => ? = 2 - 1
[[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [1,2,3,5,4,7,8,6] => ? => ? => ? = 2 - 1
[[[.,[[.,[.,.]],.]],.],[[.,.],.]]
=> [3,2,4,1,5,7,8,6] => [3,2,4,1,5,7,6] => [4,3,5,2,6,1,7] => ? = 2 - 1
[[[.,[[[.,.],.],.]],.],[[.,.],.]]
=> [2,3,4,1,5,7,8,6] => [2,3,4,1,5,7,6] => [3,4,5,2,6,1,7] => ? = 2 - 1
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,3,4,2,5,7,8,6] => ? => ? => ? = 2 - 1
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [2,1,4,3,5,7,8,6] => ? => ? => ? = 2 - 1
[[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [1,2,4,3,5,8,7,6] => ? => ? => ? = 2 - 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [3,2,1,4,5,7,6] => [4,3,2,5,6,1,7] => ? = 2 - 1
[[[[.,[.,[.,.]]],.],.],[[.,.],.]]
=> [3,2,1,4,5,7,8,6] => ? => ? => ? = 2 - 1
[[[[[.,[.,.]],.],.],.],[[.,.],.]]
=> [2,1,3,4,5,7,8,6] => ? => ? => ? = 2 - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000654
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 78%●distinct values known / distinct values provided: 64%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 78%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [1] => [] => ? = 0
[.,[.,.]]
=> [2,1] => [1,2] => [1] => ? = 1
[[.,.],.]
=> [1,2] => [2,1] => [1] => ? = 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 2
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,3,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [2,3,1] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [2,3,1] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [3,2,1] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 3
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,4,3,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,1] => 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,4,1,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [4,1,2,3] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 4
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [5,1,2,4,3] => [1,2,4,3] => 3
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [5,1,3,4,2] => [1,3,4,2] => 3
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [1,2,8,3,4,5,6,7] => ? => ? = 7
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => ? => ? = 2
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [2,3,5,4,6,7,8,1] => [1,8,7,6,4,5,3,2] => ? => ? = 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ? => ? = 2
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ? => ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [2,4,3,5,6,7,8,1] => [1,8,7,6,5,3,4,2] => ? => ? = 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,7,6,5,4,2,3] => ? = 2
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 2
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ? = 6
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [2,3,4,5,6,8,7,1] => [2,3,4,5,6,7,1] => ? = 6
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => [2,4,5,6,7,8,3,1] => ? => ? = 5
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,3,5,8,7,6,4,2] => [2,4,6,7,8,5,3,1] => ? => ? = 4
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,3,5,7,8,6,4,2] => [2,4,6,8,7,5,3,1] => [2,4,6,7,5,3,1] => ? = 4
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => [2,4,8,5,6,7,3,1] => ? => ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,3,5,7,6,8,4,2] => [2,4,8,6,7,5,3,1] => ? => ? = 4
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [1,3,6,5,7,8,4,2] => [2,4,8,7,5,6,3,1] => ? => ? = 3
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,3,4,5,6,8,7,2] => [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => ? = 2
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => [2,8,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 6
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,2] => [2,8,7,6,5,4,3,1] => ? => ? = 2
[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,8,7,6,5,4,3] => [3,4,5,6,7,8,2,1] => ? => ? = 5
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,2,4,8,7,6,5,3] => [3,5,6,7,8,4,2,1] => [3,5,6,7,4,2,1] => ? = 4
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,2,4,6,8,7,5,3] => [3,5,7,8,6,4,2,1] => ? => ? = 3
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,2,4,5,8,7,6,3] => [3,6,7,8,5,4,2,1] => ? => ? = 3
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,2,4,5,7,8,6,3] => [3,6,8,7,5,4,2,1] => ? => ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => [3,8,5,7,6,4,2,1] => ? => ? = 3
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,8,7,6,5,4] => [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => ? = 4
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,2,3,7,8,6,5,4] => [4,5,6,8,7,3,2,1] => ? => ? = 4
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,2,3,7,6,8,5,4] => [4,5,8,6,7,3,2,1] => ? => ? = 4
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => [4,6,7,8,5,3,2,1] => ? => ? = 3
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [4,6,8,7,5,3,2,1] => ? => ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,2,3,5,6,8,7,4] => [4,7,8,6,5,3,2,1] => ? => ? = 2
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,2,3,7,6,5,8,4] => [4,8,5,6,7,3,2,1] => ? => ? = 4
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,2,3,5,7,6,8,4] => [4,8,6,7,5,3,2,1] => ? => ? = 3
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 2
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,3,4,2,6,7,8,5] => [5,8,7,6,2,4,3,1] => [5,7,6,2,4,3,1] => ? = 2
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,8,7,6,5] => [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 3
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,2,3,4,7,8,6,5] => [5,6,8,7,4,3,2,1] => ? => ? = 3
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,2,3,4,6,8,7,5] => [5,7,8,6,4,3,2,1] => ? => ? = 2
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,2,3,4,7,6,8,5] => [5,8,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 3
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,6,7,8,5] => [5,8,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 2
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,2,3,5,4,8,7,6] => [6,7,8,4,5,3,2,1] => ? => ? = 2
[[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [1,2,3,5,4,7,8,6] => [6,8,7,4,5,3,2,1] => ? => ? = 2
[[[.,[[.,[.,.]],.]],.],[[.,.],.]]
=> [3,2,4,1,5,7,8,6] => [6,8,7,5,1,4,2,3] => ? => ? = 2
[[[.,[[[.,.],.],.]],.],[[.,.],.]]
=> [2,3,4,1,5,7,8,6] => [6,8,7,5,1,4,3,2] => ? => ? = 2
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,3,4,2,5,7,8,6] => [6,8,7,5,2,4,3,1] => ? => ? = 2
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [2,1,4,3,5,7,8,6] => [6,8,7,5,3,4,1,2] => ? => ? = 2
[[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [1,2,4,3,5,8,7,6] => [6,7,8,5,3,4,2,1] => ? => ? = 2
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000990
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 78%●distinct values known / distinct values provided: 55%
Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 78%●distinct values known / distinct values provided: 55%
Values
[.,.]
=> [1] => [] => [] => ? = 0
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 1
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 2
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [3,2,1] => 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [2,1,3] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [3,1,2] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [3,1,2] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,3,1] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [3,1,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [3,2,1,4] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [3,2,1,4] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [4,2,1,3] => 3
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [4,2,1,3] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [2,1,3,4] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [3,2,1,4] => 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [4,2,1,3] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [3,2,4,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [2,1,3,4] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [4,3,1,2] => 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [3,1,2,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [3,1,2,4] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [3,4,2,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,3,4,1] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 4
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [3,2,1,4] => 3
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1] => [4,2,1,3] => 3
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [3,2,4,5,1,6,7] => ? = 2
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [2,3,5,4,6,7,8,1] => ? => ? => ? = 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [4,3,5,6,7,2,1] => ? = 2
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [3,2,4,5,6,1,7] => ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [2,4,3,5,6,7,8,1] => ? => ? => ? = 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => ? = 2
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 2
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 6
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 6
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => ? => ? => ? = 5
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,3,5,8,7,6,4,2] => ? => ? => ? = 4
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,3,5,7,8,6,4,2] => [1,3,5,7,6,4,2] => [7,5,3,1,2,4,6] => ? = 4
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ? => ? => ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,3,5,7,6,8,4,2] => ? => ? => ? = 4
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [1,3,6,5,7,8,4,2] => [1,3,6,5,7,4,2] => [6,5,1,3,2,4,7] => ? = 3
[[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => [3,1,2,4,5,6,7] => ? = 2
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? => ? => ? = 6
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => [3,1,2,4,5,6,7] => ? = 2
[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,8,7,6,5,4,3] => ? => ? => ? = 5
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,2,4,8,7,6,5,3] => [1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 4
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,2,4,6,8,7,5,3] => ? => ? => ? = 3
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,2,4,5,8,7,6,3] => ? => ? => ? = 3
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,2,4,5,7,8,6,3] => ? => ? => ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ? => ? => ? = 3
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,8,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 4
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,2,3,7,8,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 4
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,2,3,7,6,8,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 4
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 3
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,2,3,5,6,8,7,4] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 2
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,2,3,7,6,5,8,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 4
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,2,3,5,7,6,8,4] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 3
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 2
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => [3,1,2,6,4,5,7] => ? = 2
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 3
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,2,3,4,7,8,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 3
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,2,3,4,6,8,7,5] => ? => ? => ? = 2
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 3
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 2
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,2,3,5,4,8,7,6] => ? => ? => ? = 2
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000181The number of connected components of the Hasse diagram for the poset. St001557The number of inversions of the second entry of a permutation.
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