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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000063
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Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> []
=> 1
[[]]
=> [1,0]
=> []
=> 1
[[],[]]
=> [1,0,1,0]
=> [1]
=> 2
[[[]]]
=> [1,1,0,0]
=> []
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 6
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 24
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 12
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 12
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 8
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 12
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 8
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 6
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 120
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 60
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 60
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 40
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 20
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 60
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 30
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 40
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 20
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 30
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 15
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 15
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 10
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 60
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 30
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 30
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 20
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 40
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 20
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 20
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 10
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 30
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 15
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 15
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 10
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000100
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 64%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 64%
Values
[]
=> .
=> ? => ?
=> ? = 1
[[]]
=> [.,.]
=> [1] => ([],1)
=> ? = 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[[[]]]
=> [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 24
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 12
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 12
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 12
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> 120
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 60
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 60
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 40
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 20
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 60
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 30
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 40
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 20
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 30
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 15
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 15
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 10
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 5
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 60
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 30
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 30
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 20
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 10
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 40
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 20
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 20
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 10
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 30
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 15
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 15
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 5
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 24
[[],[],[[[[],[]]]]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => ([(2,6),(3,6),(4,5),(6,4)],7)
=> ? = 84
[[],[],[[[[[]]]]]]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 42
[[],[[[[],[]]]],[]]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [7,3,2,4,5,6,1] => ([(2,6),(3,6),(4,5),(6,4)],7)
=> ? = 84
[[],[[[[[]]]]],[]]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 42
[[],[[],[[],[[]]]]]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[],[[],[[[]],[]]]]
=> [.,[[.,[[[.,.],[.,.]],.]],.]]
=> [5,3,4,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[],[[],[[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => ([(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> ? = 70
[[],[[],[[[[]]]]]]
=> [.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ? = 35
[[],[[[]],[[[]]]]]
=> [.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 70
[[],[[[[]]],[[]]]]
=> [.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 70
[[],[[[],[[]]],[]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[],[[[[]],[]],[]]]
=> [.,[[[[.,.],[.,.]],[.,.]],.]]
=> [6,4,2,3,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[],[[[[],[]]],[]]]
=> [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [6,3,2,4,5,7,1] => ([(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> ? = 70
[[],[[[[[]]]],[]]]
=> [.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ? = 35
[[],[[[],[],[],[]]]]
=> [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => ([(1,6),(2,6),(3,6),(4,6),(6,5)],7)
=> ? = 168
[[],[[[],[],[[]]]]]
=> [.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 84
[[],[[[],[[]],[]]]]
=> [.,[[[.,[[.,.],[.,.]]],.],.]]
=> [5,3,4,2,6,7,1] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 84
[[],[[[],[[],[]]]]]
=> [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
=> ? = 56
[[],[[[],[[[]]]]]]
=> [.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 28
[[],[[[[]],[],[]]]]
=> [.,[[[[.,.],[.,[.,.]]],.],.]]
=> [5,4,2,3,6,7,1] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 84
[[],[[[[]],[[]]]]]
=> [.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 42
[[],[[[[],[]],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
=> ? = 56
[[],[[[[[]]],[]]]]
=> [.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 28
[[],[[[[],[],[]]]]]
=> [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
=> ? = 42
[[],[[[[],[[]]]]]]
=> [.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 21
[[],[[[[[]],[]]]]]
=> [.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 21
[[],[[[[[],[]]]]]]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ? = 14
[[],[[[[[[]]]]]]]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 7
[[[]],[[[],[[]]]]]
=> [[.,.],[[[.,[[.,.],.]],.],.]]
=> [4,5,3,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 63
[[[]],[[[[]],[]]]]
=> [[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 63
[[[]],[[[[],[]]]]]
=> [[.,.],[[[[.,[.,.]],.],.],.]]
=> [4,3,5,6,7,1,2] => ([(0,6),(1,6),(2,3),(4,5),(6,4)],7)
=> ? = 42
[[[]],[[[[[]]]]]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 21
[[[[]]],[[[[]]]]]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 35
[[[[[]]]],[[[]]]]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 35
[[[[[],[]]]],[],[]]
=> [[[[.,[.,.]],.],.],[.,[.,.]]]
=> [7,6,2,1,3,4,5] => ([(2,6),(3,6),(4,5),(6,4)],7)
=> ? = 84
[[[[[[]]]]],[],[]]
=> [[[[[.,.],.],.],.],[.,[.,.]]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 42
[[[[],[[]]]],[[]]]
=> [[[.,[[.,.],.]],.],[[.,.],.]]
=> [6,7,2,3,1,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 63
[[[[[]],[]]],[[]]]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 63
[[[[[],[]]]],[[]]]
=> [[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => ([(0,6),(1,6),(2,3),(4,5),(6,4)],7)
=> ? = 42
[[[[[[]]]]],[[]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 21
[[[],[[],[[]]]],[]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[[],[[[]],[]]],[]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[[],[[[],[]]]],[]]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> [7,3,2,4,5,1,6] => ([(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> ? = 70
[[[],[[[[]]]]],[]]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [7,2,3,4,5,1,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ? = 35
[[[[]],[[[]]]],[]]
=> [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 70
[[[[[]]],[[]]],[]]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 70
[[[[],[[]]],[]],[]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> [7,5,2,3,1,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
[[[[[]],[]],[]],[]]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 105
Description
The number of linear extensions of a poset.
Matching statistic: St000110
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 84%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 84%
Values
[]
=> .
=> ? => ? = 1
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 24
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 12
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 12
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 12
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 120
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 60
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 60
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 40
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 20
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 60
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 30
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 40
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 20
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 30
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 15
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 15
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 10
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 5
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 60
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 30
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 30
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 20
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 10
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 40
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 20
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 20
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 10
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 30
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 15
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 15
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 5
[[],[],[[[[],[]]]]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => ? = 84
[[],[],[[[[[]]]]]]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ? = 42
[[],[[[[],[]]]],[]]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [7,3,2,4,5,6,1] => ? = 84
[[],[[[[[]]]]],[]]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => ? = 42
[[],[[],[[],[[]]]]]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => ? = 105
[[],[[],[[[]],[]]]]
=> [.,[[.,[[[.,.],[.,.]],.]],.]]
=> [5,3,4,6,2,7,1] => ? = 105
[[],[[],[[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => ? = 70
[[],[[[]],[[[]]]]]
=> [.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => ? = 70
[[],[[[[]]],[[]]]]
=> [.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => ? = 70
[[],[[[],[[]]],[]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 105
[[],[[[[]],[]],[]]]
=> [.,[[[[.,.],[.,.]],[.,.]],.]]
=> [6,4,2,3,5,7,1] => ? = 105
[[],[[[[],[]]],[]]]
=> [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [6,3,2,4,5,7,1] => ? = 70
[[],[[[],[],[],[]]]]
=> [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => ? = 168
[[],[[[],[],[[]]]]]
=> [.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => ? = 84
[[],[[[],[[]],[]]]]
=> [.,[[[.,[[.,.],[.,.]]],.],.]]
=> [5,3,4,2,6,7,1] => ? = 84
[[],[[[[]],[],[]]]]
=> [.,[[[[.,.],[.,[.,.]]],.],.]]
=> [5,4,2,3,6,7,1] => ? = 84
[[[]],[[[],[[]]]]]
=> [[.,.],[[[.,[[.,.],.]],.],.]]
=> [4,5,3,6,7,1,2] => ? = 63
[[[]],[[[[]],[]]]]
=> [[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => ? = 63
[[[]],[[[[],[]]]]]
=> [[.,.],[[[[.,[.,.]],.],.],.]]
=> [4,3,5,6,7,1,2] => ? = 42
[[[[]]],[[[[]]]]]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => ? = 35
[[[[[],[]]]],[],[]]
=> [[[[.,[.,.]],.],.],[.,[.,.]]]
=> [7,6,2,1,3,4,5] => ? = 84
[[[[],[[]]]],[[]]]
=> [[[.,[[.,.],.]],.],[[.,.],.]]
=> [6,7,2,3,1,4,5] => ? = 63
[[[[[]],[]]],[[]]]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => ? = 63
[[[[[],[]]]],[[]]]
=> [[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => ? = 42
[[[],[[],[[]]]],[]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => ? = 105
[[[],[[[]],[]]],[]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => ? = 105
[[[],[[[],[]]]],[]]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> [7,3,2,4,5,1,6] => ? = 70
[[[[]],[[[]]]],[]]
=> [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => ? = 70
[[[[[]]],[[]]],[]]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => ? = 70
[[[[],[[]]],[]],[]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> [7,5,2,3,1,4,6] => ? = 105
[[[[[]],[]],[]],[]]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ? = 105
[[[[[],[]]],[]],[]]
=> [[[[.,[.,.]],.],[.,.]],[.,.]]
=> [7,5,2,1,3,4,6] => ? = 70
[[[[],[],[],[]]],[]]
=> [[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [7,4,3,2,1,5,6] => ? = 168
[[[[],[],[[]]]],[]]
=> [[[.,[.,[[.,.],.]]],.],[.,.]]
=> [7,3,4,2,1,5,6] => ? = 84
[[[[],[[]],[]]],[]]
=> [[[.,[[.,.],[.,.]]],.],[.,.]]
=> [7,4,2,3,1,5,6] => ? = 84
[[[[[]],[],[]]],[]]
=> [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [7,4,3,1,2,5,6] => ? = 84
[[[],[],[],[[]],[]]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => ? = 360
[[[],[],[[]],[[]]]]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => ? = 180
[[[],[],[[[]],[]]]]
=> [[.,[.,[[[.,.],[.,.]],.]]],.]
=> [5,3,4,6,2,1,7] => ? = 90
[[[],[],[[[],[]]]]]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> [4,3,5,6,2,1,7] => ? = 60
[[[],[],[[[[]]]]]]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> [3,4,5,6,2,1,7] => ? = 30
[[[],[[]],[],[],[]]]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => ? = 360
[[[],[[]],[],[[]]]]
=> [[.,[[.,.],[.,[[.,.],.]]]],.]
=> [5,6,4,2,3,1,7] => ? = 180
[[[],[[]],[[]],[]]]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [6,4,5,2,3,1,7] => ? = 180
[[[],[[]],[[],[]]]]
=> [[.,[[.,.],[[.,[.,.]],.]]],.]
=> [5,4,6,2,3,1,7] => ? = 120
[[[],[[]],[[[]]]]]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> [4,5,6,2,3,1,7] => ? = 60
[[[],[[],[]],[],[]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [6,5,3,2,4,1,7] => ? = 240
[[[],[[[]]],[],[]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> [6,5,2,3,4,1,7] => ? = 120
[[[],[[],[]],[[]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> [5,6,3,2,4,1,7] => ? = 120
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: St000085
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000085: Ordered trees ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 84%
Values
[]
=> ? = 1
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 6
[[],[[]]]
=> 3
[[[]],[]]
=> 3
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 24
[[],[],[[]]]
=> 12
[[],[[]],[]]
=> 12
[[],[[],[]]]
=> 8
[[],[[[]]]]
=> 4
[[[]],[],[]]
=> 12
[[[]],[[]]]
=> 6
[[[],[]],[]]
=> 8
[[[[]]],[]]
=> 4
[[[],[],[]]]
=> 6
[[[],[[]]]]
=> 3
[[[[]],[]]]
=> 3
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 120
[[],[],[],[[]]]
=> 60
[[],[],[[]],[]]
=> 60
[[],[],[[],[]]]
=> 40
[[],[],[[[]]]]
=> 20
[[],[[]],[],[]]
=> 60
[[],[[]],[[]]]
=> 30
[[],[[],[]],[]]
=> 40
[[],[[[]]],[]]
=> 20
[[],[[],[],[]]]
=> 30
[[],[[],[[]]]]
=> 15
[[],[[[]],[]]]
=> 15
[[],[[[],[]]]]
=> 10
[[],[[[[]]]]]
=> 5
[[[]],[],[],[]]
=> 60
[[[]],[],[[]]]
=> 30
[[[]],[[]],[]]
=> 30
[[[]],[[],[]]]
=> 20
[[[]],[[[]]]]
=> 10
[[[],[]],[],[]]
=> 40
[[[[]]],[],[]]
=> 20
[[[],[]],[[]]]
=> 20
[[[[]]],[[]]]
=> 10
[[[],[],[]],[]]
=> 30
[[[],[[]]],[]]
=> 15
[[[[]],[]],[]]
=> 15
[[[[],[]]],[]]
=> 10
[[[[[]]]],[]]
=> 5
[[],[],[[[[],[]]]]]
=> ? = 84
[[],[],[[[[[]]]]]]
=> ? = 42
[[],[[[[],[]]]],[]]
=> ? = 84
[[],[[[[[]]]]],[]]
=> ? = 42
[[],[[],[[],[[]]]]]
=> ? = 105
[[],[[],[[[]],[]]]]
=> ? = 105
[[],[[],[[[],[]]]]]
=> ? = 70
[[],[[[]],[[[]]]]]
=> ? = 70
[[],[[[[]]],[[]]]]
=> ? = 70
[[],[[[],[[]]],[]]]
=> ? = 105
[[],[[[[]],[]],[]]]
=> ? = 105
[[],[[[[],[]]],[]]]
=> ? = 70
[[],[[[],[],[],[]]]]
=> ? = 168
[[],[[[],[],[[]]]]]
=> ? = 84
[[],[[[],[[]],[]]]]
=> ? = 84
[[],[[[[]],[],[]]]]
=> ? = 84
[[[]],[[[],[[]]]]]
=> ? = 63
[[[]],[[[[]],[]]]]
=> ? = 63
[[[]],[[[[],[]]]]]
=> ? = 42
[[[[]]],[[[[]]]]]
=> ? = 35
[[[[[]]]],[[[]]]]
=> ? = 35
[[[[[],[]]]],[],[]]
=> ? = 84
[[[[[[]]]]],[],[]]
=> ? = 42
[[[[],[[]]]],[[]]]
=> ? = 63
[[[[[]],[]]],[[]]]
=> ? = 63
[[[[[],[]]]],[[]]]
=> ? = 42
[[[],[[],[[]]]],[]]
=> ? = 105
[[[],[[[]],[]]],[]]
=> ? = 105
[[[],[[[],[]]]],[]]
=> ? = 70
[[[[]],[[[]]]],[]]
=> ? = 70
[[[[[]]],[[]]],[]]
=> ? = 70
[[[[],[[]]],[]],[]]
=> ? = 105
[[[[[]],[]],[]],[]]
=> ? = 105
[[[[[],[]]],[]],[]]
=> ? = 70
[[[[],[],[],[]]],[]]
=> ? = 168
[[[[],[],[[]]]],[]]
=> ? = 84
[[[[],[[]],[]]],[]]
=> ? = 84
[[[[[]],[],[]]],[]]
=> ? = 84
[[[],[],[],[],[],[]]]
=> ? = 720
[[[],[],[],[],[[]]]]
=> ? = 360
[[[],[],[],[[]],[]]]
=> ? = 360
[[[],[],[],[[],[]]]]
=> ? = 240
[[[],[],[],[[[]]]]]
=> ? = 120
[[[],[],[[]],[],[]]]
=> ? = 360
[[[],[],[[]],[[]]]]
=> ? = 180
[[[],[],[[],[]],[]]]
=> ? = 240
[[[],[],[[[]]],[]]]
=> ? = 120
[[[],[],[[],[],[]]]]
=> ? = 180
[[[],[],[[],[[]]]]]
=> ? = 90
Description
The number of linear extensions of the tree.
We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree $T$ on $n$ vertices, the number of linear extensions is
$$
\frac{n!}{\prod_{v\in T}|T_v|},
$$
where $T_v$ is the number of vertices of the subtree rooted at $v$.
Matching statistic: St000228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 18%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 18%
Values
[]
=> ([],1)
=> [1]
=> 1
[[]]
=> ([(0,1)],2)
=> [1]
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 24
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> ? = 12
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> ? = 12
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> ? = 12
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 120
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 60
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 60
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [10,10,10,10]
=> ? = 40
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,5,5,5]
=> ? = 20
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 60
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,5,5,5,5,5]
=> ? = 30
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [10,10,10,10]
=> ? = 40
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,5,5,5]
=> ? = 20
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [15,15]
=> ? = 30
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 15
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 15
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 10
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 5
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 60
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,5,5,5,5,5]
=> ? = 30
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,5,5,5,5,5]
=> ? = 30
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [10,10]
=> ? = 20
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> 10
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [10,10,10,10]
=> ? = 40
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,5,5,5]
=> ? = 20
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [10,10]
=> ? = 20
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> 10
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [15,15]
=> ? = 30
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 15
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 15
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 5
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [4,4,4,4,4,4]
=> ? = 24
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [4,4,4]
=> ? = 12
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [4,4,4]
=> ? = 12
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 8
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 4
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [4,4,4]
=> ? = 12
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 6
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 8
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 4
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [3,3]
=> 6
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 720
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 360
[[],[],[],[[]],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 360
[[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
=> ? = 240
[[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 120
[[],[],[[]],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 360
[[],[],[[]],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 180
[[],[],[[],[]],[]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
=> ? = 240
[[],[],[[[]]],[]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 120
[[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
=> [18,18,18,18,18,18,18,18,18,18]
=> ? = 180
[[],[],[[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> [18,18,18,18,18]
=> ? = 90
[[],[],[[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> [18,18,18,18,18]
=> ? = 90
[[],[],[[[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> [6,6,6,6,6,6,6,6,6,6]
=> ? = 60
[[],[],[[[[]]]]]
=> ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
=> [6,6,6,6,6]
=> ? = 30
[[],[[]],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 360
[[],[[]],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 180
[[],[[]],[[]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 180
[[],[[]],[[],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> [12,12,12,12,12,12,12,12,12,12]
=> ? = 120
[[],[[]],[[[]]]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> [6,6,6,6,6,6,6,6,6,6]
=> ? = 60
[[],[[],[]],[],[]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
=> ? = 240
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [6]
=> 6
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [6]
=> 6
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [5,5]
=> 10
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> 5
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> 10
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> 10
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [5,5]
=> 10
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> 5
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [8]
=> 8
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> [4]
=> 4
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> [4,2]
=> 6
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [8]
=> 8
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> [4]
=> 4
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [3,3]
=> 6
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [3]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000071
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> 24
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> 12
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> 12
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> 12
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 120
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 60
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 60
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 40
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 20
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 60
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 30
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 40
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 20
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(7,1),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ([(0,3),(0,4),(0,5),(0,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(7,1),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ? = 30
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 15
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 15
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 5
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 60
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 30
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 30
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 20
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> 10
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ?
=> ? = 40
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 20
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 20
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> 10
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(7,1),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ([(0,3),(0,4),(0,5),(0,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(7,1),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ? = 30
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 15
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 15
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 5
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,1),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,6)],18)
=> ([(0,2),(0,3),(0,4),(0,5),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,1),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,6)],18)
=> ? = 24
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 12
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 12
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> 4
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 12
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> 6
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 8
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> 4
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 6
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 720
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 360
[[],[],[],[[]],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 360
[[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 240
[[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 120
[[],[],[[]],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 360
[[],[],[[]],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 180
[[],[],[[],[]],[]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 240
[[],[],[[[]]],[]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 120
[[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 180
[[],[],[[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 90
[[],[],[[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 90
[[],[],[[[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> ?
=> ?
=> ? = 60
[[],[],[[[[]]]]]
=> ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
=> ?
=> ?
=> ? = 30
[[],[[]],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 360
[[],[[]],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 180
[[],[[]],[[]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 180
[[],[[]],[[],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 120
[[],[[]],[[[]]]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 60
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> 15
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> 20
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> 15
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> 1
Description
The number of maximal chains in a poset.
Matching statistic: St001855
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-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: 18%
Mp00014: Binary trees —to 132-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: 18%
Values
[]
=> .
=> ? => ? => ? = 1
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 24
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 12
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 12
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 4
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => 12
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ? = 120
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => ? = 60
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => ? = 60
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => ? = 40
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => ? = 20
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => ? = 60
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => ? = 30
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => ? = 40
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => ? = 20
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 30
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => ? = 15
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,5,1] => ? = 15
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 10
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 5
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => ? = 60
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => ? = 30
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 30
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 20
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? = 10
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => ? = 40
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 20
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => ? = 20
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 10
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => ? = 30
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,3,1,4] => ? = 15
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 15
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 5
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 24
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => ? = 12
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1,5] => ? = 12
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 8
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 4
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 12
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 6
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 8
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 4
[[[[],[],[]]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 6
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 3
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 3
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 720
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => ? = 360
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [6,4,5,3,2,1] => ? = 360
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => ? = 240
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => ? = 120
[[],[],[[]],[],[]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [6,5,3,4,2,1] => ? = 360
[[],[],[[]],[[]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [5,6,3,4,2,1] => ? = 180
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [6,4,3,5,2,1] => ? = 240
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
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