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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001300
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 0
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [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)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[.,[[[.,[.,.]],.],.]]
=> [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)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1
[[.,.],[[.,.],[.,.]]]
=> [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)
=> 1
[[.,.],[[[.,.],.],.]]
=> [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)
=> 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 0
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000147
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> []
=> ?
=> ? = 0
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> []
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 3
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 3
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,1,1,1]
=> ?
=> ? = 4
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,1,1,1]
=> ?
=> ? = 5
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,1,1,1]
=> ?
=> ? = 5
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,1,1,1]
=> ?
=> ? = 5
[[[.,.],.],[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,2,2,2,1]
=> ?
=> ? = 4
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3]
=> ?
=> ? = 4
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3]
=> ?
=> ? = 3
[[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3]
=> ?
=> ? = 5
[[[.,[.,.]],.],[[.,.],[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2]
=> ?
=> ? = 4
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2]
=> ?
=> ? = 5
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2]
=> [5,4,3,3,2]
=> ? = 5
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ? = 3
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ? = 3
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ?
=> ? = 5
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> [5,3,3,3,2,1]
=> ? = 5
[[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4]
=> ?
=> ? = 5
[[.,[.,[[.,.],.]]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1]
=> ?
=> ? = 5
[[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1]
=> ?
=> ? = 4
[[.,[[.,.],[.,.]]],[[[.,.],.],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1]
=> ?
=> ? = 5
[[.,[[.,[.,.]],.]],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2]
=> ?
=> ? = 5
[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ?
=> ? = 4
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1,1]
=> ?
=> ? = 5
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,2,2]
=> ?
=> ? = 4
[[[.,[.,.]],[.,.]],[[[.,.],.],.]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2,2]
=> ?
=> ? = 5
[[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3]
=> ?
=> ? = 4
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3]
=> ?
=> ? = 5
[[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2]
=> ?
=> ? = 4
[[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2]
=> [5,4,4,3,2]
=> ? = 5
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,2,2]
=> ?
=> ? = 5
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,2,2,1]
=> ?
=> ? = 5
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2]
=> ?
=> ? = 5
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,2]
=> ?
=> ? = 5
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2,1]
=> ?
=> ? = 5
[[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,2,1]
=> ?
=> ? = 5
[[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4]
=> ?
=> ? = 5
[[[.,[[.,[.,.]],.]],.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2]
=> ?
=> ? = 5
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,2]
=> ?
=> ? = 5
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ?
=> ? = 5
[[[[[.,[.,.]],.],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 5
[[[.,.],[[[.,[.,.]],.],.]],[.,.]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,1,1,1]
=> ?
=> ? = 6
[[[.,[[.,.],.]],[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3,1]
=> ?
=> ? = 6
[[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,3,2]
=> ?
=> ? = 6
[[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3,2,1]
=> ?
=> ? = 6
Description
The largest part of an integer partition.
Matching statistic: St000734
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [[1]]
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [[1]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [[1,3],[2]]
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [[1,3],[2]]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => ? => ?
=> ? = 3 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [4,3,7,6,5,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [3,4,7,6,5,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? => ?
=> ? = 1 + 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [3,2,8,7,6,5,4,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => ? => ?
=> ? = 4 + 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => ? => ?
=> ? = 4 + 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [2,3,8,7,6,5,4,1] => ? => ?
=> ? = 2 + 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => ? => ?
=> ? = 4 + 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [3,6,5,4,7,2,8,1] => ? => ?
=> ? = 4 + 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [2,7,6,5,4,3,8,1] => ? => ?
=> ? = 1 + 1
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [3,2,7,6,5,4,8,1] => ? => ?
=> ? = 2 + 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [2,3,7,6,5,4,8,1] => ? => ?
=> ? = 2 + 1
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,6,5,7,8,4,3,2] => ? => ?
=> ? = 3 + 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => ? => ?
=> ? = 2 + 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ? => ?
=> ? = 2 + 1
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,5,4,3,7,8,6,2] => ? => ?
=> ? = 4 + 1
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [1,4,3,5,7,8,6,2] => ? => ?
=> ? = 4 + 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ? => ?
=> ? = 5 + 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? => ?
=> ? = 1 + 1
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [1,6,5,7,4,3,8,2] => ? => ?
=> ? = 3 + 1
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [1,5,4,6,7,3,8,2] => ? => ?
=> ? = 4 + 1
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [1,5,4,6,3,7,8,2] => ? => ?
=> ? = 5 + 1
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [1,4,3,6,5,7,8,2] => ? => ?
=> ? = 5 + 1
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [2,1,7,6,8,5,4,3] => ? => ?
=> ? = 2 + 1
[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,8,7,6,5,4,3] => ? => ?
=> ? = 2 + 1
[[[.,.],.],[.,[[[.,[.,.]],.],.]]]
=> [1,2,6,5,7,8,4,3] => ? => ?
=> ? = 4 + 1
[[[.,[.,.]],.],[.,[.,[[.,.],.]]]]
=> [2,1,3,7,8,6,5,4] => ? => ?
=> ? = 3 + 1
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
=> [2,1,3,6,7,8,5,4] => ? => ?
=> ? = 4 + 1
[[[.,[.,.]],.],[[.,.],[[.,.],.]]]
=> [2,1,3,5,7,8,6,4] => ? => ?
=> ? = 4 + 1
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [2,1,3,6,5,7,8,4] => ? => ?
=> ? = 5 + 1
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
=> [2,1,3,5,6,7,8,4] => ? => ?
=> ? = 5 + 1
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [2,1,5,4,3,7,8,6] => ? => ?
=> ? = 5 + 1
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,2,5,4,3,7,8,6] => ? => ?
=> ? = 5 + 1
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [2,1,3,5,4,8,7,6] => ? => ?
=> ? = 5 + 1
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [2,1,3,5,4,7,8,6] => ? => ?
=> ? = 5 + 1
[[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,2,3,5,4,8,7,6] => ? => ?
=> ? = 5 + 1
[[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [1,2,3,5,4,7,8,6] => ? => ?
=> ? = 5 + 1
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [2,1,4,3,5,7,8,6] => ? => ?
=> ? = 5 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000957
(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
St000957: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 0
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [2,1,3,4] => 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1] => [1,3,2,4] => 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 2
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 2
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 2
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 3
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 3
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 4
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => ? = 4
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 4
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => ? = 4
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 2
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [2,3,4,5,1,6,7] => ? = 4
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 4
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 4
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [3,4,5,6,2,1,7] => ? = 5
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [4,5,6,3,2,1,7] => ? = 5
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [5,6,4,3,2,1,7] => ? = 5
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,3,1,2] => [6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => ? = 2
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,3,1,2] => [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ? = 3
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [2,3,1,5,4,7,6] => ? = 4
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [4,5,3,2,1,7,6] => ? = 5
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [6,5,7,4,3,8,1,2] => [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ? = 3
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [3,4,2,1,5,7,6] => ? = 4
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [3,4,2,5,1,7,6] => ? = 5
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [2,4,5,3,1,7,6] => ? = 5
[[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> [5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [3,4,5,2,1,7,6] => ? = 5
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [4,5,3,2,1,7,6] => ? = 5
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,2,1,3] => [5,6,7,4,2,1,3] => [3,2,1,4,6,7,5] => ? = 4
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,2,1,3] => [4,5,6,7,2,1,3] => [4,3,2,1,6,7,5] => ? = 5
[[[.,.],.],[.,[[[.,[.,.]],.],.]]]
=> [6,5,7,8,4,1,2,3] => [6,5,7,4,1,2,3] => [2,3,1,4,7,6,5] => ? = 4
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [6,5,4,7,8,1,2,3] => [6,5,4,7,1,2,3] => [2,3,4,1,7,6,5] => ? = 5
[[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> [5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,4] => [3,2,1,5,6,7,4] => ? = 5
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,2,1,3,4] => [6,7,5,2,1,3,4] => [2,1,3,6,7,5,4] => ? = 4
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,2,1,3,4] => [6,5,7,2,1,3,4] => [2,3,1,6,7,5,4] => ? = 5
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,2,1,3,4] => [5,6,7,2,1,3,4] => [3,2,1,6,7,5,4] => ? = 5
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,1,2,3,4] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 5
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,1,2,3,4] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 5
[[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> [6,7,8,4,3,2,1,5] => [6,7,4,3,2,1,5] => [2,1,4,5,6,7,3] => ? = 5
[[.,[.,[[.,.],.]]],[[[.,.],.],.]]
=> [6,7,8,3,4,2,1,5] => [6,7,3,4,2,1,5] => [2,1,5,4,6,7,3] => ? = 5
[[.,[[.,.],[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,2,3,1,5] => [6,7,4,2,3,1,5] => [2,1,4,6,5,7,3] => ? = 5
[[.,[[.,[.,.]],.]],[[[.,.],.],.]]
=> [6,7,8,3,2,4,1,5] => [6,7,3,2,4,1,5] => [2,1,5,6,4,7,3] => ? = 5
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,3,1,2,5] => [6,7,4,3,1,2,5] => [2,1,4,5,7,6,3] => ? = 5
[[[.,[.,.]],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,2,1,3,5] => [6,7,4,2,1,3,5] => [2,1,4,6,7,5,3] => ? = 5
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [6,7,8,3,2,1,4,5] => [6,7,3,2,1,4,5] => [2,1,5,6,7,4,3] => ? = 5
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000019
(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
St000019: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 0
[[.,.],.]
=> [1,2] => [1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 2
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [7,6,4,5,8,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => [1,3,4,2,5,6,7] => ? = 2
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [7,4,5,6,8,3,2,1] => [7,4,5,6,3,2,1] => [1,4,3,2,5,6,7] => ? = 2
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 2
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,3,4,5,2,6,7] => ? = 3
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 3
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [7,6,5,3,4,8,2,1] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => [1,2,4,5,3,6,7] => ? = 2
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 2
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 4
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 2
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => [1,3,2,4,6,5,7] => ? = 2
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => [1,2,4,3,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,3,5,6,4,7] => ? = 2
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => ? = 4
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 4
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,2,3,4,1] => [7,6,5,2,3,4,1] => [1,2,3,6,5,4,7] => ? = 2
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => ? = 4
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 4
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 4
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [7,6,5,3,2,4,8,1] => [7,6,5,3,2,4,1] => [1,2,3,5,6,4,7] => ? = 2
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [7,6,5,2,3,4,8,1] => [7,6,5,2,3,4,1] => [1,2,3,6,5,4,7] => ? = 2
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [3,4,5,6,2,1,7] => ? = 5
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,3,1,2] => [6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => ? = 2
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,3,1,2] => [7,5,6,4,3,1,2] => [1,3,2,4,5,7,6] => ? = 2
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,3,1,2] => [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ? = 3
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [7,8,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ? = 2
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ? = 2
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ? = 2
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [2,3,1,5,4,7,6] => ? = 4
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => [1,3,4,5,2,7,6] => ? = 4
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => [1,4,5,3,2,7,6] => ? = 4
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [4,5,3,2,1,7,6] => ? = 5
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [6,5,7,4,3,8,1,2] => [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ? = 3
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [3,4,2,1,5,7,6] => ? = 4
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [3,4,2,5,1,7,6] => ? = 5
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [2,4,5,3,1,7,6] => ? = 5
[[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> [5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [3,4,5,2,1,7,6] => ? = 5
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [4,5,3,2,1,7,6] => ? = 5
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => ? = 2
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => ? = 2
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,2,1,3] => [5,6,7,4,2,1,3] => [3,2,1,4,6,7,5] => ? = 4
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,2,1,3] => [4,5,6,7,2,1,3] => [4,3,2,1,6,7,5] => ? = 5
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St001280
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> []
=> ? = 0
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1]
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> []
=> ? = 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[[.,[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 0
[[[[.,.],.],.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 4
[[[[.,.],.],[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ? = 4
[[[[.,.],[.,.]],.],[[.,.],.]]
=> [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 4
[[[[.,[.,.]],.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> ? = 4
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 4
[[[[[.,.],.],.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 4
[[[[.,.],.],[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 5
[[[[.,.],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ? = 5
[[[[.,[.,.]],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ? = 5
[[[[[.,.],.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 5
[[[[.,.],[[.,.],.]],.],[.,.]]
=> [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 5
[[[[.,[.,.]],[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 5
[[[[[.,.],.],[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 5
[[[[.,[[.,.],.]],.],.],[.,.]]
=> [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 5
[[[[[.,.],[.,.]],.],.],[.,.]]
=> [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 5
[[[[[.,[.,.]],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 5
[[[[[[.,.],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 5
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 4
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 4
[[[[.,[.,.]],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 4
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 3
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 4
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 4
[[[[.,[.,.]],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 4
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 4
[[[[.,[[.,.],.]],.],[.,.]],.]
=> [.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 4
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 4
[[[[[.,[.,.]],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 4
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 4
[[[[.,.],[[[.,.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 5
[[[[.,[.,.]],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 5
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 5
[[[[[.,.],.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 5
[[[[.,[[.,.],.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 5
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 5
[[[[[.,[.,.]],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 5
[[[[[[.,.],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 5
[[[[.,[[[.,.],.],.]],.],.],.]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 5
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 5
[[[[[.,.],[[.,.],.]],.],.],.]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 5
[[[[[.,[.,.]],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 5
[[[[[[.,.],.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 5
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 5
[[[[[.,[[.,.],.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 5
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000653
(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
St000653: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,2] => [1] => ? = 0
[[.,.],.]
=> [1,0,1,0]
=> [2,1] => [1] => ? = 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 3
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 3
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => ? => ? = 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] => ? => ? = 0
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [6,1,8,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 2
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7] => ? = 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [5,8,1,2,3,4,6,7] => ? => ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6,8] => [5,7,1,2,3,4,6] => ? = 2
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [5,1,2,8,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [5,6,1,8,2,3,4,7] => ? => ? = 2
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 3
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7,8] => [4,1,2,3,5,6,7] => ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [4,8,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6,8] => [4,7,1,2,3,5,6] => ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,6,1,2,3,5,7,8] => [4,6,1,2,3,5,7] => ? = 2
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,2,6,3,4,5,7,8] => ? => ? = 3
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,6,7,2,8,3,4,5] => [1,6,7,2,3,4,5] => ? = 3
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4,1,2,3,8,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,5,2,3,8,4,6,7] => ? => ? = 2
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,5,1,2,8,3,6,7] => [4,5,1,2,3,6,7] => ? = 2
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,2,3,4] => [1,5,6,7,2,3,4] => ? = 4
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7] => ? = 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? => ? = 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6,8] => [3,7,1,2,4,5,6] => ? = 2
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,6,1,2,4,5,7,8] => ? => ? = 2
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,5,1,2,4,6,7,8] => ? => ? = 2
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,6,7,8,2,3,5] => ? => ? = 4
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7] => ? = 2
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,4,5,6,8,2,3,7] => ? => ? = 4
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? = 2
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,2,7,3,8,4,5] => [1,6,2,7,3,4,5] => ? = 4
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [4,1,2,7,3,8,5,6] => ? => ? = 4
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,1,2,4,5,8,6,7] => [3,1,2,4,5,6,7] => ? = 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,4,1,2,5,8,6,7] => ? => ? = 2
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,7,8,4,5] => ? => ? = 5
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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] => ? = 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] => ? = 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => ? = 2
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => ? = 2
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,1,7,8,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 3
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => ? => ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ? => ? = 2
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,3,5,6,7,8] => ? => ? = 2
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [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] => ? = 2
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000141
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => 0
[[.,.],.]
=> [1,2] => [2,1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [2,3,1] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [2,3,1] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [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] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 1
[.,[[[.,[.,.]],.],.]]
=> [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] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [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] => 1
[[.,.],[[[.,.],.],.]]
=> [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] => 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => [1,2,3,4,6,8,7,5] => ? => ? = 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => [1,2,3,4,8,6,7,5] => [1,2,3,4,6,7,5] => ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => ? => ? = 2
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,5] => ? = 2
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => ? = 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => [1,2,3,5,6,8,7,4] => [1,2,3,5,6,7,4] => ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => [1,2,3,5,7,8,6,4] => [1,2,3,5,7,6,4] => ? = 2
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 2
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [1,2,3,8,5,6,7,4] => [1,2,3,5,6,7,4] => ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => [1,2,3,8,6,7,4,5] => ? => ? = 2
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => [1,2,3,8,6,7,5,4] => [1,2,3,6,7,5,4] => ? = 2
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [1,2,3,8,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => [1,2,4,5,6,8,7,3] => ? => ? = 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => [1,2,4,5,7,8,6,3] => ? => ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [1,2,4,6,7,8,5,3] => [1,2,4,6,7,5,3] => ? = 2
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => [1,2,6,7,8,3,4,5] => ? => ? = 3
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [1,2,8,3,4,5,6,7] => ? => ? = 0
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [1,2,8,3,7,6,4,5] => [1,2,3,7,6,4,5] => ? = 3
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => [1,2,8,4,5,6,7,3] => ? => ? = 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [4,3,7,6,5,8,2,1] => [1,2,8,5,6,7,3,4] => ? => ? = 2
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [3,4,7,6,5,8,2,1] => [1,2,8,5,6,7,4,3] => ? => ? = 2
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [1,2,8,7,6,5,3,4] => ? => ? = 4
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => ? = 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => ? = 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [1,3,4,5,7,6,2] => ? = 2
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => [1,3,4,6,7,8,5,2] => ? => ? = 2
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [1,3,5,6,7,8,4,2] => [1,3,5,6,7,4,2] => ? = 2
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [3,2,8,7,6,5,4,1] => [1,4,5,6,7,8,2,3] => [1,4,5,6,7,2,3] => ? = 2
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => [1,4,7,8,5,6,2,3] => ? => ? = 4
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => [1,4,8,7,6,5,2,3] => ? => ? = 4
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [2,3,8,7,6,5,4,1] => [1,4,5,6,7,8,3,2] => ? => ? = 2
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => [1,6,8,7,5,4,2,3] => ? => ? = 4
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 2
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 4
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => ? = 4
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [3,6,5,4,7,2,8,1] => [1,8,2,7,4,5,6,3] => ? => ? = 4
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [2,7,6,5,4,3,8,1] => [1,8,3,4,5,6,7,2] => ? => ? = 1
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [3,2,7,6,5,4,8,1] => [1,8,4,5,6,7,2,3] => ? => ? = 2
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [2,3,7,6,5,4,8,1] => [1,8,4,5,6,7,3,2] => ? => ? = 2
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [1,8,7,6,2,3,4,5] => [1,7,6,2,3,4,5] => ? = 5
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ? => ? = 5
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,7,6,5,4,2,3] => ? = 5
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,6,8,7,5,4,3,2] => [2,3,4,5,7,8,6,1] => ? => ? = 2
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,5,8,7,6,4,3,2] => [2,3,4,6,7,8,5,1] => ? => ? = 2
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,6,5,7,8,4,3,2] => [2,3,4,8,7,5,6,1] => [2,3,4,7,5,6,1] => ? = 3
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,4,8,7,6,5,3,2] => [2,3,5,6,7,8,4,1] => [2,3,5,6,7,4,1] => ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,4,7,8,6,5,3,2] => [2,3,5,6,8,7,4,1] => [2,3,5,6,7,4,1] => ? = 2
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => [2,4,5,6,7,8,3,1] => ? => ? = 2
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => [2,4,8,5,6,7,3,1] => ? => ? = 2
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: St000740
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [4,1,3,2] => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [3,2,4,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [3,4,1,2] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [3,4,1,2] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [2,4,1,3] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [3,2,4,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [3,4,1,2] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,4,1,3] => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [2,3,1,4] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [3,2,4,1] => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [4,5,3,7,1,6,2] => ? = 1 + 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [4,7,1,6,2,5,3] => ? = 2 + 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [4,6,2,7,1,5,3] => ? = 2 + 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => [7,1,4,5,3,6,2] => ? = 1 + 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => ? => ? => ? = 2 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [4,7,1,6,2,5,3] => ? = 2 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [4,7,6,5,3,2,1] => [7,1,4,5,3,6,2] => ? = 1 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => ? => ? => ? = 2 + 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => ? => ? => ? = 2 + 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [6,2,5,3,7,1,4] => ? = 3 + 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => ? => ? => ? = 1 + 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => ? => ? => ? = 1 + 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => ? => ? => ? = 2 + 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => [6,2,5,4,7,1,3] => ? = 2 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => ? => ? => ? = 3 + 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [7,1,5,3,6,2,4] => ? = 3 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => ? => ? => ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [4,3,7,6,5,8,2,1] => ? => ? => ? = 2 + 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [3,4,7,6,5,8,2,1] => ? => ? => ? = 2 + 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [7,1,4,6,2,3,5] => ? = 4 + 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? => ? => ? = 1 + 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => [5,4,6,3,7,1,2] => ? = 1 + 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [2,6,7,5,4,3,1] => [5,4,7,1,2,6,3] => ? = 2 + 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => ? => ? => ? = 2 + 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [2,4,7,6,5,3,1] => [5,7,1,2,4,6,3] => ? = 2 + 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [3,2,8,7,6,5,4,1] => ? => ? => ? = 2 + 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => ? => ? => ? = 4 + 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [2,3,8,7,6,5,4,1] => ? => ? => ? = 2 + 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0 + 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [4,5,3,7,1,6,2] => ? = 1 + 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [4,7,1,6,2,5,3] => ? = 2 + 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [4,3,7,1,6,2,5] => ? = 4 + 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [6,2,4,3,7,1,5] => ? = 4 + 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [3,6,5,4,7,2,8,1] => ? => ? => ? = 4 + 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [2,7,6,5,4,3,8,1] => ? => ? => ? = 1 + 1
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [3,2,7,6,5,4,8,1] => ? => ? => ? = 2 + 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [2,3,7,6,5,4,8,1] => ? => ? => ? = 2 + 1
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [3,4,2,7,1,5,6] => ? = 5 + 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,7,1,4,5,6] => ? = 5 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000727
(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
Mp00066: Permutations —inverse⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [4,2,1,3] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,2,1,3] => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 1 + 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 2 + 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 2 + 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 1 + 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => ? => ? => ? = 2 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 2 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 1 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => ? => ? => ? = 2 + 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => ? => ? => ? = 2 + 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 3 + 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => ? => ? => ? = 1 + 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => ? => ? => ? = 1 + 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => ? => ? => ? = 2 + 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => ? => ? => ? = 3 + 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 3 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => ? => ? => ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [4,3,7,6,5,8,2,1] => ? => ? => ? = 2 + 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [3,4,7,6,5,8,2,1] => ? => ? => ? = 2 + 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => ? = 4 + 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? => ? => ? = 1 + 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => [7,1,6,5,4,3,2] => ? = 1 + 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [2,6,7,5,4,3,1] => [7,1,6,5,4,2,3] => ? = 2 + 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => ? => ? => ? = 2 + 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [2,4,7,6,5,3,1] => [7,1,6,2,5,4,3] => ? = 2 + 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [3,2,8,7,6,5,4,1] => ? => ? => ? = 2 + 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => ? => ? => ? = 4 + 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [2,3,8,7,6,5,4,1] => ? => ? => ? = 2 + 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 1 + 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 2 + 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => ? = 4 + 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 4 + 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [3,6,5,4,7,2,8,1] => ? => ? => ? = 4 + 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [2,7,6,5,4,3,8,1] => ? => ? => ? = 1 + 1
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [3,2,7,6,5,4,8,1] => ? => ? => ? = 2 + 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [2,3,7,6,5,4,8,1] => ? => ? => ? = 2 + 1
Description
The largest label of a leaf in the binary search tree associated with the permutation.
Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St001557The number of inversions of the second entry of a permutation.
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