Your data matches 20 different statistics following compositions of up to 3 maps.
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Mp00061: Permutations to increasing treeBinary trees
St000396: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1
[1,2] => [.,[.,.]]
=> 1
[2,1] => [[.,.],.]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2
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.
Mp00061: Permutations to increasing treeBinary trees
Mp00141: Binary trees pruning number to logarithmic heightDyck paths
Mp00121: Dyck paths Cori-Le Borgne involutionDyck paths
St000920: Dyck paths ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,5,7,6,8,4,3,2] => [.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[1,5,4,7,8,6,3,2] => [.,[[[[.,.],[[.,[.,.]],.]],.],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,4,6,5,8,7,3,2] => [.,[[[.,[[.,.],[[.,.],.]]],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
[1,3,5,6,8,7,4,2] => [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,3,6,5,7,8,4,2] => [.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[1,4,3,7,8,6,5,2] => [.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,4,3,6,8,7,5,2] => [.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,4,3,6,7,8,5,2] => [.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[1,3,4,7,6,8,5,2] => [.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[1,3,5,4,8,7,6,2] => [.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,4,3,5,7,8,6,2] => [.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 2
[1,4,6,5,3,8,7,2] => [.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,5,4,3,6,8,7,2] => [.,[[[[.,.],.],[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[1,3,4,5,7,6,8,2] => [.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[1,3,4,6,5,7,8,2] => [.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[2,1,6,8,7,5,4,3] => [[.,.],[[[[.,[[.,.],.]],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[2,1,5,8,7,6,4,3] => [[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[2,1,5,7,8,6,4,3] => [[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[2,1,5,6,8,7,4,3] => [[.,.],[[[.,[.,[[.,.],.]]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[2,1,7,6,5,8,4,3] => [[.,.],[[[[[.,.],.],[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[2,1,5,7,6,8,4,3] => [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 2
[2,1,6,5,7,8,4,3] => [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[2,1,4,8,7,6,5,3] => [[.,.],[[.,[[[[.,.],.],.],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[2,1,4,7,8,6,5,3] => [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[2,1,4,6,8,7,5,3] => [[.,.],[[.,[[.,[[.,.],.]],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[2,1,4,6,7,8,5,3] => [[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[2,1,4,5,8,7,6,3] => [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[2,1,4,5,7,8,6,3] => [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[2,1,6,5,4,8,7,3] => [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
[2,1,4,6,5,8,7,3] => [[.,.],[[.,[[.,.],[[.,.],.]]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[2,1,5,4,6,8,7,3] => [[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[2,1,4,5,6,8,7,3] => [[.,.],[[.,[.,[.,[[.,.],.]]]],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[2,1,6,7,5,4,8,3] => [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
[2,1,5,7,6,4,8,3] => [[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[2,1,4,7,6,5,8,3] => [[.,.],[[.,[[[.,.],.],[.,.]]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[2,1,5,4,6,7,8,3] => [[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,2,6,5,8,7,4,3] => [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,5,7,6,8,4,3] => [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[1,2,5,4,7,8,6,3] => [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,2,4,5,7,8,6,3] => [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,2,4,6,5,8,7,3] => [.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,5,4,6,8,7,3] => [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
[1,2,5,4,6,7,8,3] => [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,3,2,6,8,7,5,4] => [.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,3,2,7,6,8,5,4] => [.,[[.,.],[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,3,2,6,7,8,5,4] => [.,[[.,.],[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,3,2,5,8,7,6,4] => [.,[[.,.],[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,3,2,5,7,8,6,4] => [.,[[.,.],[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,3,2,6,5,8,7,4] => [.,[[.,.],[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[1,3,2,5,6,8,7,4] => [.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Mp00071: Permutations descent compositionInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000264: Graphs ⟶ ℤResult quality: 33% values known / values provided: 78%distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1 + 1
[1,2] => [2] => [1] => ([],1)
=> ? = 1 + 1
[2,1] => [1,1] => [2] => ([],2)
=> ? = 1 + 1
[1,2,3] => [3] => [1] => ([],1)
=> ? = 1 + 1
[1,3,2] => [2,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[2,1,3] => [1,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[3,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? = 1 + 1
[1,2,3,4] => [4] => [1] => ([],1)
=> ? = 1 + 1
[1,2,4,3] => [3,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,4] => [2,2] => [2] => ([],2)
=> ? = 2 + 1
[1,3,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[1,4,2,3] => [2,2] => [2] => ([],2)
=> ? = 2 + 1
[1,4,3,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[2,1,3,4] => [1,3] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[2,1,4,3] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => [2] => ([],2)
=> ? = 2 + 1
[2,3,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[2,4,1,3] => [2,2] => [2] => ([],2)
=> ? = 2 + 1
[2,4,3,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[3,1,2,4] => [1,3] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[3,1,4,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[3,2,4,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => [2] => ([],2)
=> ? = 2 + 1
[3,4,2,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[4,1,2,3] => [1,3] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[4,1,3,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[4,2,3,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[4,3,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 1
[1,2,3,4,5] => [5] => [1] => ([],1)
=> ? = 1 + 1
[1,2,3,5,4] => [4,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[1,2,4,3,5] => [3,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,2,4,5,3] => [4,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[1,2,5,3,4] => [3,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,2,5,4,3] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,3,2,5,4] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[1,3,4,2,5] => [3,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,3,4,5,2] => [4,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[1,3,5,2,4] => [3,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,3,5,4,2] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,4,2,5,3] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[1,4,5,2,3] => [3,2] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,4,5,3,2] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[1,5,2,3,4] => [2,3] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[1,5,2,4,3] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[1,5,3,2,4] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,3,4,2] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[1,5,4,2,3] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,4,3,2] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[2,1,3,4,5] => [1,4] => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
[2,1,3,5,4] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,4,5,3] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,5,4,3] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1,5] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,3,1,4] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,1,3] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,5,4] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,4,5,2] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,5,4,2] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,5,4] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,5,1] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,5,4,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1,5] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,2,1,4] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,4,1,2] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,5,3] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,3,5,2] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,5,3,2] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,5,3] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,5,1] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,5,3,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,5,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,5,1] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,5,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,5,2,1,3] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,5,3,1,2] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,2,4,3] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,3,4,2] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,4,3,2] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,2,1,4,3] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,2,3,4,1] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,4,3,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,1,4,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,2,4,1] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,4,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,4,1,3,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,4,2,3,1] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,3,6,5] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,3,6,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,4,3,6] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,4,6,3] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,6,3,5,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
Mp00247: Graphs de-duplicateGraphs
St001734: Graphs ⟶ ℤResult quality: 38% values known / values provided: 38%distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[2,1,4,3,6,5,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,4,3,7,5,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,4,3,7,6,5] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,3,6,4,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,3,7,4,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,3,7,6,4] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,4,3,7,6] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,4,6,3,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,4,7,3,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,5,4,7,6,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,3,5,4,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,3,7,4,5] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,3,7,5,4] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,4,3,7,5] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,4,5,3,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,4,7,3,5] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,4,7,5,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,5,3,7,4] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,5,4,3,7] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,5,4,7,3] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,5,7,3,4] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,6,5,7,4,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,3,5,4,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,3,6,4,5] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,3,6,5,4] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,4,3,6,5] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,4,5,3,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,4,6,3,5] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,4,6,5,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,5,3,6,4] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,5,4,3,6] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,5,4,6,3] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,5,6,3,4] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,5,6,4,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,6,3,5,4] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,6,4,3,5] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,6,4,5,3] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,6,5,3,4] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,4,2,6,5,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,4,2,7,5,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,4,2,7,6,5] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,2,6,4,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,2,7,4,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,2,7,6,4] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,4,2,7,6] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,4,6,2,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,4,7,2,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,4,7,6,2] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,6,2,5,4,7] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
Description
The lettericity of a graph. Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$. The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.
Mp00061: Permutations to increasing treeBinary trees
Mp00013: Binary trees to posetPosets
Mp00198: Posets incomparability graphGraphs
St000455: Graphs ⟶ ℤResult quality: 18% values known / values provided: 18%distinct values known / distinct values provided: 33%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 2 - 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
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.
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00295: Standard tableaux valley compositionInteger compositions
St001235: Integer compositions ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 67%
Values
[1] => [[1]]
=> [1] => 1
[1,2] => [[1,2]]
=> [2] => 1
[2,1] => [[1],[2]]
=> [2] => 1
[1,2,3] => [[1,2,3]]
=> [3] => 1
[1,3,2] => [[1,2],[3]]
=> [3] => 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => 2
[2,3,1] => [[1,2],[3]]
=> [3] => 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => 2
[1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7]]
=> [7] => ? = 1
[1,2,3,4,5,7,6] => [[1,2,3,4,5,6],[7]]
=> [7] => ? = 1
[1,2,3,4,6,5,7] => [[1,2,3,4,5,7],[6]]
=> [6,1] => ? = 2
[1,2,3,4,6,7,5] => [[1,2,3,4,5,6],[7]]
=> [7] => ? = 1
[1,2,3,4,7,5,6] => [[1,2,3,4,5,7],[6]]
=> [6,1] => ? = 2
[1,2,3,4,7,6,5] => [[1,2,3,4,5],[6],[7]]
=> [7] => ? = 1
[1,2,3,5,4,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,2] => ? = 2
[1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,2] => ? = 2
[1,2,3,5,6,4,7] => [[1,2,3,4,5,7],[6]]
=> [6,1] => ? = 2
[1,2,3,5,6,7,4] => [[1,2,3,4,5,6],[7]]
=> [7] => ? = 1
[1,2,3,5,7,4,6] => [[1,2,3,4,5],[6,7]]
=> [6,1] => ? = 2
[1,2,3,5,7,6,4] => [[1,2,3,4,5],[6],[7]]
=> [7] => ? = 1
[1,2,3,6,4,5,7] => [[1,2,3,4,6,7],[5]]
=> [5,2] => ? = 2
[1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,2] => ? = 2
[1,2,3,6,5,4,7] => [[1,2,3,4,7],[5],[6]]
=> [6,1] => ? = 2
[1,2,3,6,5,7,4] => [[1,2,3,4,6],[5],[7]]
=> [5,2] => ? = 2
[1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,1] => ? = 2
[1,2,3,6,7,5,4] => [[1,2,3,4,5],[6],[7]]
=> [7] => ? = 1
[1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,2] => ? = 2
[1,2,3,7,4,6,5] => [[1,2,3,4,6],[5],[7]]
=> [5,2] => ? = 2
[1,2,3,7,5,4,6] => [[1,2,3,4,7],[5],[6]]
=> [6,1] => ? = 2
[1,2,3,7,5,6,4] => [[1,2,3,4,6],[5],[7]]
=> [5,2] => ? = 2
[1,2,3,7,6,4,5] => [[1,2,3,4,7],[5],[6]]
=> [6,1] => ? = 2
[1,2,3,7,6,5,4] => [[1,2,3,4],[5],[6],[7]]
=> [7] => ? = 1
[1,2,4,3,5,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,3] => ? = 2
[1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,3] => ? = 2
[1,2,4,3,6,5,7] => [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ? = 2
[1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,3] => ? = 2
[1,2,4,3,7,5,6] => [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ? = 2
[1,2,4,3,7,6,5] => [[1,2,3,5],[4,6],[7]]
=> [4,3] => ? = 2
[1,2,4,5,3,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,2] => ? = 2
[1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,2] => ? = 2
[1,2,4,5,6,3,7] => [[1,2,3,4,5,7],[6]]
=> [6,1] => ? = 2
[1,2,4,5,6,7,3] => [[1,2,3,4,5,6],[7]]
=> [7] => ? = 1
[1,2,4,5,7,3,6] => [[1,2,3,4,5],[6,7]]
=> [6,1] => ? = 2
[1,2,4,5,7,6,3] => [[1,2,3,4,5],[6],[7]]
=> [7] => ? = 1
[1,2,4,6,3,5,7] => [[1,2,3,4,7],[5,6]]
=> [5,2] => ? = 2
[1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,2] => ? = 2
[1,2,4,6,5,3,7] => [[1,2,3,4,7],[5],[6]]
=> [6,1] => ? = 2
[1,2,4,6,5,7,3] => [[1,2,3,4,6],[5],[7]]
=> [5,2] => ? = 2
[1,2,4,6,7,3,5] => [[1,2,3,4,5],[6,7]]
=> [6,1] => ? = 2
[1,2,4,6,7,5,3] => [[1,2,3,4,5],[6],[7]]
=> [7] => ? = 1
[1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,2] => ? = 2
[1,2,4,7,3,6,5] => [[1,2,3,4],[5,6],[7]]
=> [5,2] => ? = 2
[1,2,4,7,5,3,6] => [[1,2,3,4,7],[5],[6]]
=> [6,1] => ? = 2
[1,2,4,7,5,6,3] => [[1,2,3,4,6],[5],[7]]
=> [5,2] => ? = 2
[1,2,4,7,6,3,5] => [[1,2,3,4],[5,7],[6]]
=> [6,1] => ? = 2
[1,2,4,7,6,5,3] => [[1,2,3,4],[5],[6],[7]]
=> [7] => ? = 1
[1,2,5,3,4,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,3] => ? = 2
[1,2,5,3,4,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,3] => ? = 2
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Mp00061: Permutations to increasing treeBinary trees
Mp00008: Binary trees to complete treeOrdered trees
St000397: Ordered trees ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 67%
Values
[1] => [.,.]
=> [[],[]]
=> 2 = 1 + 1
[1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[1,3,2] => [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 2 + 1
[2,3,1] => [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 2 + 1
[3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 2 = 1 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> 2 = 1 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 3 = 2 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 3 = 2 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 2 = 1 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 3 = 2 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 3 = 2 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 3 = 2 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 3 = 2 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 2 = 1 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 2 = 1 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> 3 = 2 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> 3 = 2 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> 2 = 1 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> 3 = 2 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> 2 = 1 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> 3 = 2 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 3 = 2 + 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[],[[],[[],[[],[[],[[],[[],[]]]]]]]]
=> ? = 1 + 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[],[[],[[],[[],[[],[[[],[]],[]]]]]]]
=> ? = 1 + 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[[],[[],[[],[[[],[]],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[],[[],[[],[[],[[[],[[],[]]],[]]]]]]
=> ? = 1 + 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[[],[[],[[],[[[],[]],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [[],[[],[[],[[],[[[[],[]],[]],[]]]]]]
=> ? = 1 + 1
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ? = 2 + 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[],[[],[[],[[[],[[],[[],[]]]],[]]]]]
=> ? = 1 + 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [[],[[],[[],[[[],[[[],[]],[]]],[]]]]]
=> ? = 1 + 1
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ? = 2 + 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[],[[],[[],[[[[],[]],[[],[]]],[]]]]]
=> ? = 2 + 1
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [[],[[],[[],[[[[],[[],[]]],[]],[]]]]]
=> ? = 1 + 1
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ? = 2 + 1
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[],[[],[[],[[[[],[]],[[],[]]],[]]]]]
=> ? = 2 + 1
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [[],[[],[[],[[[[[],[]],[]],[]],[]]]]]
=> ? = 1 + 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[[],[[[],[]],[[],[[],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [[],[[],[[[],[]],[[],[[[],[]],[]]]]]]
=> ? = 2 + 1
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[[],[[[],[]],[[[],[]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[],[[],[[[],[]],[[[],[[],[]]],[]]]]]
=> ? = 2 + 1
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[[],[[[],[]],[[[],[]],[[],[]]]]]]
=> ? = 2 + 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [[],[[],[[[],[]],[[[[],[]],[]],[]]]]]
=> ? = 2 + 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ? = 2 + 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ? = 2 + 1
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ? = 2 + 1
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[],[[],[[[],[[],[[],[[],[]]]]],[]]]]
=> ? = 1 + 1
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ? = 2 + 1
[1,2,4,5,7,6,3] => [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [[],[[],[[[],[[],[[[],[]],[]]]],[]]]]
=> ? = 1 + 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ? = 2 + 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ? = 2 + 1
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ? = 2 + 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[],[[],[[[],[[[],[]],[[],[]]]],[]]]]
=> ? = 2 + 1
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ? = 2 + 1
[1,2,4,6,7,5,3] => [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [[],[[],[[[],[[[],[[],[]]],[]]],[]]]]
=> ? = 1 + 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ? = 2 + 1
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ? = 2 + 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[],[[],[[[],[[[],[]],[[],[]]]],[]]]]
=> ? = 2 + 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ? = 2 + 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [[],[[],[[[],[[[[],[]],[]],[]]],[]]]]
=> ? = 1 + 1
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[[],[[[],[]],[[],[[],[[],[]]]]]]]
=> ? = 2 + 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [[],[[],[[[],[]],[[],[[[],[]],[]]]]]]
=> ? = 2 + 1
Description
The Strahler number of a rooted tree.
Matching statistic: St000308
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00223: Permutations runsortPermutations
Mp00064: Permutations reversePermutations
St000308: Permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [1,3,2] => [2,3,1] => 2
[2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2
[1,3,4,2] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2
[1,4,3,2] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2
[2,1,4,3] => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[2,3,1,4] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,4,1,3] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[2,4,3,1] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2
[3,4,1,2] => [3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2
[4,2,1,3] => [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2
[4,3,1,2] => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,4,5,3] => [4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,3,2,5,4] => [3,5,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,3,4,2,5] => [3,4,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,3,4,5,2] => [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,5,2,4] => [5,3,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,3,5,4,2] => [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 2
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => 2
[1,4,3,2,5] => [4,3,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,4,3,5,2] => [4,3,5,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,4,5,2,3] => [4,1,5,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => [6,5,4,3,2,7,1] => ? = 2
[1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 2
[1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,5,7,6,4] => [7,5,6,1,2,3,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 2
[1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => [1,6,7,2,3,4,5] => [5,4,3,2,7,6,1] => ? = 2
[1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,6,5,7,4] => [6,5,7,1,2,3,4] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,6,7,4,5] => [6,1,7,2,3,4,5] => [1,7,2,3,4,5,6] => [6,5,4,3,2,7,1] => ? = 2
[1,2,3,6,7,5,4] => [6,7,5,1,2,3,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 2
[1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 2
[1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => [1,5,2,3,4,6,7] => [7,6,4,3,2,5,1] => ? = 2
[1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => [1,7,2,3,4,5,6] => [6,5,4,3,2,7,1] => ? = 2
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 2
[1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 2
[1,2,4,3,6,5,7] => [4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 2
[1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,3,7,5,6] => [7,4,1,2,3,5,6] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 2
[1,2,4,3,7,6,5] => [7,4,6,1,2,3,5] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,5,3,6,7] => [4,5,1,2,3,6,7] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 2
[1,2,4,5,3,7,6] => [4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 2
[1,2,4,5,6,3,7] => [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,7,3,6] => [7,4,5,1,2,3,6] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 2
[1,2,4,5,7,6,3] => [7,4,5,6,1,2,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,6,3,5,7] => [6,4,1,2,3,5,7] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 2
[1,2,4,6,3,7,5] => [6,4,7,1,2,3,5] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 2
[1,2,4,6,5,3,7] => [6,4,5,1,2,3,7] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,6,5,7,3] => [6,4,5,7,1,2,3] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,4,6,7,3,5] => [6,7,4,1,2,3,5] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,6,7,5,3] => [6,7,4,5,1,2,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,7,3,5,6] => [7,1,4,2,3,5,6] => [1,4,2,3,5,6,7] => [7,6,5,3,2,4,1] => ? = 2
[1,2,4,7,3,6,5] => [4,7,6,1,2,3,5] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 2
[1,2,4,7,5,3,6] => [4,7,5,1,2,3,6] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 2
[1,2,4,7,5,6,3] => [4,7,5,6,1,2,3] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,4,7,6,3,5] => [7,6,4,1,2,3,5] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,7,6,5,3] => [7,6,4,5,1,2,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,5,3,4,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => [7,6,4,3,2,5,1] => ? = 2
[1,2,5,3,4,7,6] => [1,5,7,2,3,4,6] => [1,5,7,2,3,4,6] => [6,4,3,2,7,5,1] => ? = 2
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Mp00126: Permutations cactus evacuationPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St001174: Permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[2,1,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,3,4,2] => [3,1,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[2,4,3,1] => [4,2,1,3] => [2,1,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,3,4,2] => [3,4,1,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => 1 = 2 - 1
[3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,4,3,1] => [4,2,3,1] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,4,3,1] => 1 = 2 - 1
[4,3,1,2] => [1,4,3,2] => [4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [4,1,2,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,4,2,5] => [3,4,1,2,5] => 1 = 2 - 1
[1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,3,4,5,2] => [3,1,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [3,5,1,2,4] => [1,5,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [4,1,5,2,3] => [4,1,2,5,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 1 = 2 - 1
[1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => 1 = 2 - 1
[1,4,5,2,3] => [4,5,1,2,3] => [1,4,2,5,3] => 1 = 2 - 1
[1,4,5,3,2] => [4,3,1,2,5] => [1,4,3,2,5] => 0 = 1 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => ? = 1 - 1
[1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 2 - 1
[1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => ? = 1 - 1
[1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ? = 2 - 1
[1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? = 1 - 1
[1,2,3,5,4,6,7] => [1,2,5,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2 - 1
[1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => [1,2,5,3,7,4,6] => ? = 2 - 1
[1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 2 - 1
[1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => [1,2,3,5,4,6,7] => ? = 1 - 1
[1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => [1,2,3,7,5,4,6] => ? = 2 - 1
[1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => [1,2,3,5,4,7,6] => ? = 1 - 1
[1,2,3,6,4,5,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => ? = 2 - 1
[1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => [1,2,6,3,4,7,5] => ? = 2 - 1
[1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => [1,2,6,5,3,4,7] => ? = 2 - 1
[1,2,3,6,5,7,4] => [6,1,5,2,3,4,7] => [1,2,6,3,5,4,7] => ? = 2 - 1
[1,2,3,6,7,4,5] => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => ? = 2 - 1
[1,2,3,6,7,5,4] => [6,5,1,2,3,4,7] => [1,2,3,6,5,4,7] => ? = 1 - 1
[1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => [1,2,3,7,4,5,6] => ? = 2 - 1
[1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => [1,2,3,7,4,6,5] => ? = 2 - 1
[1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => [1,2,7,3,5,4,6] => ? = 2 - 1
[1,2,3,7,5,6,4] => [7,1,5,2,3,4,6] => [1,2,5,3,4,7,6] => ? = 2 - 1
[1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => [1,2,3,7,6,4,5] => ? = 2 - 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? = 1 - 1
[1,2,4,3,5,6,7] => [1,2,4,5,3,6,7] => [4,5,1,2,3,6,7] => ? = 2 - 1
[1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => [1,4,2,7,3,5,6] => ? = 2 - 1
[1,2,4,3,6,5,7] => [1,4,2,6,3,5,7] => [4,1,6,2,3,5,7] => ? = 2 - 1
[1,2,4,3,6,7,5] => [4,1,2,6,3,5,7] => [4,1,2,6,3,5,7] => ? = 2 - 1
[1,2,4,3,7,5,6] => [1,4,2,7,3,5,6] => [1,7,4,2,3,5,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => [1,4,2,7,3,6,5] => ? = 2 - 1
[1,2,4,5,3,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 2 - 1
[1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => [1,2,7,4,3,5,6] => ? = 2 - 1
[1,2,4,5,6,3,7] => [1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => ? = 2 - 1
[1,2,4,5,6,7,3] => [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1 - 1
[1,2,4,5,7,3,6] => [4,7,1,2,3,5,6] => [1,2,4,3,7,5,6] => ? = 2 - 1
[1,2,4,5,7,6,3] => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => ? = 1 - 1
[1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,6,4,2,3,5,7] => ? = 2 - 1
[1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [1,4,2,6,3,5,7] => ? = 2 - 1
[1,2,4,6,5,3,7] => [1,6,4,2,3,5,7] => [1,6,2,4,3,5,7] => ? = 2 - 1
[1,2,4,6,5,7,3] => [6,1,4,2,3,5,7] => [1,4,2,3,6,5,7] => ? = 2 - 1
[1,2,4,6,7,3,5] => [4,6,1,2,3,5,7] => [1,2,6,4,3,5,7] => ? = 2 - 1
[1,2,4,6,7,5,3] => [6,4,1,2,3,5,7] => [1,2,4,3,6,5,7] => ? = 1 - 1
[1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 2 - 1
[1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [1,2,7,4,3,6,5] => ? = 2 - 1
[1,2,4,7,5,3,6] => [1,7,4,2,3,5,6] => [1,4,2,3,7,5,6] => ? = 2 - 1
[1,2,4,7,5,6,3] => [7,1,4,2,3,5,6] => [1,4,2,3,5,7,6] => ? = 2 - 1
[1,2,4,7,6,3,5] => [4,7,6,1,2,3,5] => [1,2,7,6,4,3,5] => ? = 2 - 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [1,2,4,3,7,6,5] => ? = 1 - 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00223: Permutations runsortPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St001859: Permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[1,3,4,2] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [3,4,1,2] => 1 = 2 - 1
[2,1,4,3] => [2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[2,3,1,4] => [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[2,4,3,1] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [1,3,4,2] => [3,4,1,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[3,4,1,2] => [3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[4,2,1,3] => [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,4,5,3] => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,2,4,5,3] => [4,5,1,2,3] => 1 = 2 - 1
[1,3,2,5,4] => [3,5,1,2,4] => [1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,3,4,2,5] => [3,4,1,2,5] => [1,2,5,3,4] => [1,5,2,3,4] => 1 = 2 - 1
[1,3,4,5,2] => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [5,3,1,2,4] => [1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,3,5,4,2] => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,2,5,3,4] => [1,5,2,3,4] => 1 = 2 - 1
[1,4,3,5,2] => [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 2 - 1
[1,4,5,2,3] => [4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,5,3,2] => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ? = 2 - 1
[1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 2 - 1
[1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 2 - 1
[1,2,3,5,7,6,4] => [7,5,6,1,2,3,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 2 - 1
[1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => [1,6,7,2,3,4,5] => [1,2,6,3,7,4,5] => ? = 2 - 1
[1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,2,3,6,5,7,4] => [6,5,7,1,2,3,4] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,6,7,4,5] => [6,1,7,2,3,4,5] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ? = 2 - 1
[1,2,3,6,7,5,4] => [6,7,5,1,2,3,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [1,2,3,7,4,5,6] => ? = 2 - 1
[1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 2 - 1
[1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => [1,5,2,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 2 - 1
[1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ? = 2 - 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[1,2,4,3,6,5,7] => [4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => ? = 2 - 1
[1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[1,2,4,3,7,5,6] => [7,4,1,2,3,5,6] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[1,2,4,3,7,6,5] => [7,4,6,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[1,2,4,5,3,6,7] => [4,5,1,2,3,6,7] => [1,2,3,6,7,4,5] => [6,1,7,2,3,4,5] => ? = 2 - 1
[1,2,4,5,3,7,6] => [4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 2 - 1
[1,2,4,5,6,3,7] => [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 2 - 1
[1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,5,7,3,6] => [7,4,5,1,2,3,6] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 2 - 1
[1,2,4,5,7,6,3] => [7,4,5,6,1,2,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,6,3,5,7] => [6,4,1,2,3,5,7] => [1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => ? = 2 - 1
[1,2,4,6,3,7,5] => [6,4,7,1,2,3,5] => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ? = 2 - 1
[1,2,4,6,5,3,7] => [6,4,5,1,2,3,7] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 2 - 1
[1,2,4,6,5,7,3] => [6,4,5,7,1,2,3] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,4,6,7,3,5] => [6,7,4,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[1,2,4,6,7,5,3] => [6,7,4,5,1,2,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,7,3,5,6] => [7,1,4,2,3,5,6] => [1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => ? = 2 - 1
[1,2,4,7,3,6,5] => [4,7,6,1,2,3,5] => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ? = 2 - 1
[1,2,4,7,5,3,6] => [4,7,5,1,2,3,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 2 - 1
[1,2,4,7,5,6,3] => [4,7,5,6,1,2,3] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[1,2,4,7,6,5,3] => [7,6,4,5,1,2,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,5,3,4,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 2 - 1
Description
The number of factors of the Stanley symmetric function associated with a permutation. For example, the Stanley symmetric function of $\pi=321645$ equals $20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000699The toughness times the least common multiple of 1,. St001823The Stasinski-Voll length of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.