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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000110
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Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 120
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 60
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 40
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 40
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 40
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 20
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 30
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 15
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 20
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 30
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 20
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 15
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 30
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 15
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 10
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 10
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 20
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000100
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 57%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 57%
Values
[1] => [.,.]
=> [1] => ([],1)
=> ? = 1
[1,2] => [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[2,1] => [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> 120
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 60
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 40
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 40
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 40
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 20
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 30
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 15
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 20
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 30
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 20
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 15
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 30
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 15
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 10
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 10
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 20
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 10
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([],7)
=> ? = 5040
[1,3,7,6,5,4,2] => [.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ? = 35
[1,4,6,7,5,3,2] => [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
=> ? = 56
[1,4,7,6,5,3,2] => [.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 28
[1,5,6,7,4,3,2] => [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
=> ? = 42
[1,5,7,6,4,3,2] => [.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 21
[1,6,7,5,4,3,2] => [.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ? = 14
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 7
[2,1,3,4,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ? = 240
[2,1,3,5,6,4,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 120
[2,1,3,5,6,7,4] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> ? = 180
[2,1,3,5,7,4,6] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 120
[2,1,3,5,7,6,4] => [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [1,5,6,4,7,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(4,6),(5,1)],7)
=> ? = 90
[2,1,3,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 120
[2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 144
[2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 240
[2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 240
[2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 240
[2,3,4,6,5,7,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [4,6,5,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 240
[2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 240
[2,3,4,7,5,6,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [4,6,5,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 240
[2,3,5,4,6,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 180
[2,3,5,4,7,6,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 90
[2,3,5,6,4,7,1] => [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 120
[2,3,5,6,7,1,4] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 240
[2,3,5,7,4,6,1] => [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 120
[2,3,6,4,5,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 180
[2,3,6,4,7,5,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 90
[2,3,6,7,4,5,1] => [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 120
[2,3,7,4,5,6,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 180
[2,3,7,4,6,5,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 90
[2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 240
[2,4,3,5,6,7,1] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 144
[2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 240
[2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 240
[2,5,3,4,6,7,1] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 144
[2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 240
[2,6,3,4,5,7,1] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 144
[2,6,5,4,3,1,7] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 10
[2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 240
[2,7,3,4,5,6,1] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 144
[2,7,5,4,3,1,6] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 10
[2,7,6,4,3,1,5] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 10
[2,7,6,5,3,1,4] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 10
[2,7,6,5,4,1,3] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 10
[3,1,2,4,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ? = 240
[3,1,2,5,6,4,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 120
[3,1,2,5,6,7,4] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> ? = 180
[3,1,2,5,7,4,6] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 120
Description
The number of linear extensions of a poset.
Matching statistic: St001346
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 47%
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 47%
Values
[1] => [.,.]
=> [.,.]
=> [1] => ? = 1
[1,2] => [.,[.,.]]
=> [[.,.],.]
=> [1,2] => 2
[2,1] => [[.,.],.]
=> [.,[.,.]]
=> [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => 6
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 3
[2,1,3] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[3,1,2] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [3,2,1] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 120
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 60
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 40
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 40
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 40
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 20
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 30
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 15
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 20
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 30
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 20
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 15
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 30
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 15
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 10
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 10
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 20
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 10
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ? = 5040
[1,3,7,6,5,4,2] => [.,[[.,[[[[.,.],.],.],.]],.]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [5,4,3,2,6,1,7] => ? = 35
[1,4,6,7,5,3,2] => [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => ? = 56
[1,4,7,6,5,3,2] => [.,[[[.,[[[.,.],.],.]],.],.]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => ? = 28
[1,5,6,7,4,3,2] => [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => ? = 42
[1,5,7,6,4,3,2] => [.,[[[[.,[[.,.],.]],.],.],.]]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => ? = 21
[1,6,7,5,4,3,2] => [.,[[[[[.,[.,.]],.],.],.],.]]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => ? = 14
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => ? = 7
[2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => ? = 720
[2,1,3,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> [7,2,1,3,4,5,6] => ? = 360
[2,1,3,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => ? = 240
[2,1,3,4,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> [7,2,3,1,4,5,6] => ? = 240
[2,1,3,4,7,5,6] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => ? = 240
[2,1,3,4,7,6,5] => [[.,.],[.,[.,[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [7,3,2,1,4,5,6] => ? = 120
[2,1,3,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => ? = 180
[2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [7,4,2,1,3,5,6] => ? = 90
[2,1,3,5,6,4,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ? = 120
[2,1,3,5,6,7,4] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [[[.,[[[.,.],.],.]],.],[.,.]]
=> [7,2,3,4,1,5,6] => ? = 180
[2,1,3,5,7,4,6] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ? = 120
[2,1,3,5,7,6,4] => [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [[[.,[[.,[.,.]],.]],.],[.,.]]
=> [7,3,2,4,1,5,6] => ? = 90
[2,1,3,6,4,5,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => ? = 180
[2,1,3,6,4,7,5] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [7,4,2,1,3,5,6] => ? = 90
[2,1,3,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ? = 120
[2,1,3,7,4,5,6] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => ? = 180
[2,1,3,7,4,6,5] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [7,4,2,1,3,5,6] => ? = 90
[2,1,4,3,5,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => ? = 144
[2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [7,2,3,4,5,1,6] => ? = 144
[2,1,5,3,4,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => ? = 144
[2,1,6,3,4,5,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => ? = 144
[2,1,7,3,4,5,6] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => ? = 144
[2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ? = 240
[2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ? = 240
[2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 720
[2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ? = 240
[2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => ? = 360
[2,3,4,6,5,7,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => ? = 240
[2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ? = 240
[2,3,4,6,7,5,1] => [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => ? = 240
[2,3,4,7,5,6,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => ? = 240
[2,3,4,7,6,5,1] => [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => ? = 120
[2,3,5,4,6,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => ? = 180
[2,3,5,4,7,6,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => ? = 90
[2,3,5,6,4,7,1] => [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => ? = 120
[2,3,5,6,7,1,4] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ? = 240
[2,3,5,6,7,4,1] => [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => ? = 180
[2,3,5,7,4,6,1] => [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => ? = 120
[2,3,5,7,6,4,1] => [[.,[.,[[.,[[.,.],.]],.]]],.]
=> [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => ? = 90
[2,3,6,4,5,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => ? = 180
[2,3,6,4,7,5,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => ? = 90
Description
The number of parking functions that give the same permutation.
A '''parking function''' $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
Matching statistic: St001855
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 13%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 13%
Values
[1] => [.,.]
=> [1] => [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ? = 120
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => ? = 60
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => ? = 40
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => ? = 40
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => ? = 40
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => ? = 20
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? = 30
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,5,3,1] => ? = 15
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 20
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 30
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 20
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => ? = 15
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? = 30
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,5,3,1] => ? = 15
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? = 10
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,5,1] => ? = 10
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 20
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 10
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? = 30
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,5,3,1] => ? = 15
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? = 10
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,5,1] => ? = 10
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? = 10
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 5
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 24
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,3,2] => 12
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,5,2] => 8
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => 4
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 12
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? = 12
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 24
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? = 12
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => ? = 12
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 12
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? = 8
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? = 12
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 8
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 12
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? = 8
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 4
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 24
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,3,2] => 12
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,5,2] => 8
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => 4
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 6
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,5,3] => 3
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 4
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,3,2,5] => 6
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 4
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 12
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? = 12
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 6
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 12
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
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