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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000535
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => ([],1)
=> 0
[.,[.,.]]
=> [2,1] => [1,2] => ([],2)
=> 0
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => ([],3)
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => ([],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => ([],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => ([],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => ([],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => ([],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => ([],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => ([],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => ([],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => ([],5)
=> 0
Description
The rank-width of a graph.
Matching statistic: St000396
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1 = 0 + 1
[[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,[.,.]],.],.],.]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000862
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [3,1,2,4] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [4,1,2,3,5] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,3,2,4,5] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 1 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,2,3,1] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 1 + 1
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [7,4,3,5,2,6,1] => [1,2,6,3,5,4,7] => [1,6,2,5,3,4,7] => ? = 1 + 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [7,5,2,3,4,6,1] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [7,4,3,2,5,6,1] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => ? = 1 + 1
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [7,3,4,2,5,6,1] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => ? = 1 + 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [7,4,2,3,5,6,1] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 1 + 1
[.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [1,2,7,3,4,6,5] => [7,1,2,6,3,4,5] => ? = 1 + 1
[.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [6,4,3,5,2,7,1] => [1,2,7,3,5,4,6] => [1,7,2,5,3,4,6] => ? = 1 + 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [1,2,7,3,6,4,5] => [1,7,2,6,3,4,5] => ? = 1 + 1
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [1,2,7,3,6,4,5] => [1,7,2,6,3,4,5] => ? = 1 + 1
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [5,3,4,6,2,7,1] => [1,2,7,3,4,6,5] => [7,1,2,6,3,4,5] => ? = 1 + 1
[.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => [1,2,7,3,5,6,4] => [7,1,2,5,3,4,6] => ? = 1 + 1
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> [5,4,6,2,3,7,1] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 1 + 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> [6,4,2,3,5,7,1] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => ? = 1 + 1
[.,[[[[.,[.,.]],.],[.,.]],.]]
=> [6,3,2,4,5,7,1] => [1,2,4,5,7,3,6] => [4,7,1,2,3,5,6] => ? = 1 + 1
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => [1,2,6,7,3,5,4] => [6,5,7,1,2,3,4] => ? = 1 + 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [7,4,3,5,6,1,2] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 1 + 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [5,4,6,3,7,1,2] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 1 + 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => ? = 1 + 1
[[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => [1,3,2,4,5,7,6] => [3,1,2,4,7,5,6] => ? = 1 + 1
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> [6,4,5,7,2,1,3] => [1,3,2,4,5,7,6] => [3,1,2,4,7,5,6] => ? = 1 + 1
[[.,[.,.]],[[[.,[.,.]],.],.]]
=> [5,4,6,7,2,1,3] => [1,3,2,4,6,7,5] => [3,1,2,4,6,5,7] => ? = 1 + 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,1,2,3] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [6,5,7,3,2,1,4] => [1,4,2,3,5,7,6] => [4,1,2,3,7,5,6] => ? = 1 + 1
[[.,[[.,.],.]],[[.,[.,.]],.]]
=> [6,5,7,2,3,1,4] => [1,4,2,3,5,7,6] => [4,1,2,3,7,5,6] => ? = 1 + 1
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [6,5,7,3,1,2,4] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 1 + 1
[[[.,[.,.]],.],[[.,[.,.]],.]]
=> [6,5,7,2,1,3,4] => [1,3,4,2,5,7,6] => [3,1,2,7,4,5,6] => ? = 1 + 1
[[.,[.,[[.,[.,.]],.]]],[.,.]]
=> [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => ? = 1 + 1
[[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => ? = 1 + 1
[[.,[[[.,[.,.]],.],.]],[.,.]]
=> [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => [1,6,2,4,5,3,7] => ? = 1 + 1
[[[.,.],[[.,[.,.]],.]],[.,.]]
=> [7,4,3,5,1,2,6] => [1,2,6,3,5,4,7] => [1,6,2,5,3,4,7] => ? = 1 + 1
[[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 + 1
[[[.,[[.,[.,.]],.]],.],[.,.]]
=> [7,3,2,4,1,5,6] => [1,5,6,2,4,3,7] => [1,5,4,6,2,3,7] => ? = 1 + 1
[[[[.,.],[.,[.,.]]],.],[.,.]]
=> [7,4,3,1,2,5,6] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => ? = 1 + 1
[[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => ? = 1 + 1
[[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 1 + 1
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => [7,1,2,3,6,4,5] => ? = 1 + 1
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => [1,7,2,3,5,4,6] => ? = 1 + 1
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,7,2,3,6,4,5] => ? = 1 + 1
[[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,7,2,3,6,4,5] => ? = 1 + 1
[[.,[.,[[[.,.],[.,.]],.]]],.]
=> [5,3,4,6,2,1,7] => [1,7,2,3,4,6,5] => [7,1,2,3,6,4,5] => ? = 1 + 1
[[.,[.,[[[.,[.,.]],.],.]]],.]
=> [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => [7,1,2,3,5,4,6] => ? = 1 + 1
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St000485
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => ? = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,4,3,2,5] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,4,3,2,5] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,5,3,4,2] => 2 = 1 + 1
[[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[.,[[.,.],.]]],[[.,.],.]]
=> [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[[.,.],[.,.]]],[[.,.],.]]
=> [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[[.,[.,.]],.]],[.,[.,.]]]
=> [7,6,3,2,4,1,5] => [1,5,2,4,3,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[[.,[.,.]],.]],[[.,.],.]]
=> [6,7,3,2,4,1,5] => [1,5,2,4,3,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[[[.,.],.],.]],[.,[.,.]]]
=> [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[[.,[.,[.,.]]],.],[.,[.,.]]]
=> [7,6,3,2,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[[[.,[[.,.],.]],.],[.,[.,.]]]
=> [7,6,2,3,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[[[.,[[.,.],.]],.],[[.,.],.]]
=> [6,7,2,3,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[.,[[.,[.,.]],.]]],[.,.]]
=> [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[.,[[[.,.],.],.]]],[.,.]]
=> [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[[.,.],[[.,.],.]]],[.,.]]
=> [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[[.,[.,.]],[.,.]]],[.,.]]
=> [7,5,3,2,4,1,6] => [1,6,2,4,3,5,7] => [1,6,3,5,4,2,7] => ? = 1 + 1
[[.,[[[.,.],.],[.,.]]],[.,.]]
=> [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[[.,[.,[.,.]]],.]],[.,.]]
=> [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => [1,6,3,5,4,2,7] => ? = 1 + 1
[[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => [1,6,3,5,4,2,7] => ? = 1 + 1
[[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[[[.,[.,.]],.],.]],[.,.]]
=> [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => [1,6,3,5,4,2,7] => ? = 1 + 1
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[[.,[.,[.,.]]],[.,.]],[.,.]]
=> [7,5,3,2,1,4,6] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[[[.,[[.,.],.]],[.,.]],[.,.]]
=> [7,5,2,3,1,4,6] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[[[[.,[.,.]],.],[.,.]],[.,.]]
=> [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [1,5,3,6,2,4,7] => ? = 1 + 1
[[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [7,4,3,2,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[.,[.,[[.,.],.]]],.],[.,.]]
=> [7,3,4,2,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[.,[[.,.],[.,.]]],.],[.,.]]
=> [7,4,2,3,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[.,[[.,[.,.]],.]],.],[.,.]]
=> [7,3,2,4,1,5,6] => [1,5,6,2,4,3,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [7,2,3,4,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[[.,[.,.]],[.,.]],.],[.,.]]
=> [7,4,2,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [7,3,2,1,4,5,6] => [1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[[.,[[.,.],.]],.],.],[.,.]]
=> [7,2,3,1,4,5,6] => [1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[[[[[.,[.,.]],.],.],.],[.,.]]
=> [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [6,5,3,4,2,1,7] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
Description
The length of the longest cycle of a permutation.
Matching statistic: St000298
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,2,3,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [7,5,6,3,2,4,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1 + 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [7,6,4,3,2,5,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [7,6,3,4,2,5,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [7,6,3,2,4,5,1] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [7,5,4,3,2,6,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [7,4,5,3,2,6,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [7,5,3,4,2,6,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [7,4,3,5,2,6,1] => [1,2,6,3,5,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 1 + 1
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [7,3,4,5,2,6,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [7,5,4,2,3,6,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [7,4,5,2,3,6,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [7,5,3,2,4,6,1] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [7,5,2,3,4,6,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1 + 1
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [7,4,3,2,5,6,1] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ? = 1 + 1
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [7,3,4,2,5,6,1] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ? = 1 + 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [7,4,2,3,5,6,1] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [7,3,2,4,5,6,1] => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 1 + 1
[.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [6,4,3,5,2,7,1] => [1,2,7,3,5,4,6] => ([(0,5),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [6,5,3,2,4,7,1] => [1,2,4,7,3,5,6] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [1,2,4,7,3,5,6] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 1 + 1
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => [1,2,5,7,3,4,6] => ([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7)
=> ? = 1 + 1
[.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => [1,2,5,7,3,4,6] => ([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7)
=> ? = 1 + 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> [6,4,2,3,5,7,1] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 1 + 1
[.,[[[[.,[.,.]],.],[.,.]],.]]
=> [6,3,2,4,5,7,1] => [1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 1 + 1
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => [1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
=> ? = 1 + 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1 + 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [7,4,5,3,6,1,2] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1 + 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [7,4,3,5,6,1,2] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,2,1,3] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[[[.,.],.],.]]]
=> [5,6,7,4,2,1,3] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 56%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 56%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([],1)
=> ? = 0 - 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 - 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[[.,.],[[.,.],.]],.]]
=> [.,[[[.,.],[[.,.],.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[.,[[[.,[.,.]],[.,.]],.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000929
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [.,.]
=> ([],1)
=> [1]
=> ? = 0 - 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> [1]
=> ? = 0 - 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> [1]
=> ? = 0 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 0 = 1 - 1
[.,[[[[[.,.],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[[.,.],[.,[.,[[.,.],.]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0 - 1
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001568
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 53%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 53%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [.,.]
=> ([],1)
=> [1]
=> ? = 0
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> [1]
=> ? = 0
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> [1]
=> ? = 0
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? = 0
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? = 0
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? = 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 1
[.,[[[[[.,.],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
[[.,.],[.,[[.,.],[.,.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? = 0
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001624
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,6),(4,6),(5,1),(6,7),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,6),(4,6),(5,1),(6,7),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [.,[[.,[[.,[.,.]],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [.,[[.,[[.,.],[.,[.,.]]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [.,[[.,[[.,.],[[.,.],.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [.,[[.,[[[.,.],[.,.]],.]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [.,[[.,[[[.,.],.],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,7),(3,6),(4,6),(5,7),(6,2),(7,1)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [.,[[[.,[[.,.],[.,.]]],.],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(0,3),(0,4),(0,5),(2,7),(3,6),(4,6),(5,2),(6,7),(7,1)],8)
=> ? = 1 + 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([],1)
=> ? = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,6),(4,6),(5,1),(6,7),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,6),(4,6),(5,1),(6,7),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [.,[[.,[[.,[.,.]],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [.,[[.,[[.,.],[.,[.,.]]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [.,[[.,[[.,.],[[.,.],.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [.,[[.,[[[.,.],[.,.]],.]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [.,[[.,[[[.,.],.],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,7),(3,6),(4,6),(5,7),(6,2),(7,1)],8)
=> ? = 1 + 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [.,[[[.,[[.,.],[.,.]]],.],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,2),(5,1),(6,4),(7,5)],8)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(4,1),(5,2),(6,4),(7,3)],8)
=> ? = 1 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001335The cardinality of a minimal cycle-isolating set of a graph. St000397The Strahler number of a rooted tree. St000308The height of the tree associated to a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001741The largest integer such that all patterns of this size are contained in the permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St000640The rank of the largest boolean interval in a poset. St001621The number of atoms of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001823The Stasinski-Voll length of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000805The number of peaks of the associated bargraph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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