- FindStat

Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000396
(load all 10 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [.,.]
 => 1
[1,0,1,0]
 => [[.,.],.]
 => 1
[1,1,0,0]
 => [.,[.,.]]
 => 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => 1
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 2
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 1
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 2
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 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.

Matching statistic: St000920
(load all 6 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [.,.]
 => [1,0]
 => 1
[1,0,1,0]
 => [[.,.],.]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [.,[.,.]]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [[.,.],[[[[[.,.],.],.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[[[.,.],.],.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[[[.,.],.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [[.,.],[[[[.,.],.],[.,[.,.]]],.]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [[.,.],[[[.,.],.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [[.,.],[[[.,.],.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [[.,.],[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [[.,.],[[[.,.],[[.,[.,.]],.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [[.,.],[[.,.],[[[.,[.,.]],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [[.,.],[[[.,.],[.,[[.,.],.]]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [[.,.],[[[.,[.,.]],.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[[.,[.,.]],.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[[.,[.,.]],[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [[.,.],[[[.,[.,.]],[[.,.],.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [[.,.],[[.,[.,.]],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[.,[[.,.],.]],.],.],.]]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [[.,.],[[[.,[[.,.],.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [[.,.],[[[.,[[.,.],.]],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [[.,.],[[.,[[.,.],.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [[.,.],[[.,[[.,.],.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,[[[.,.],.],.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,[[[.,.],.],.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [[.,.],[[.,[[[[.,.],.],.],.]],.]]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [[.,.],[.,[[[[[.,.],.],.],.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [[.,.],[.,[[[[.,.],.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St000264

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => [1] => [1] => ([],1)
 => ? = 1 + 1
[1,0,1,0]
 => [1,1] => [2] => ([],2)
 => ? = 1 + 1
[1,1,0,0]
 => [2] => [1] => ([],1)
 => ? = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => ([],3)
 => ? = 1 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,0,0,1,0]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,0,0,0]
 => [3] => [1] => ([],1)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => ([],4)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2] => ([],2)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [2] => ([],2)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1] => ([],1)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 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.

Matching statistic: St001734

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001734: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [2] => ([],2)
 => ([],1)
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,1,0,1,0,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,1,1,0,0,0]
 => [3] => ([],3)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0]
 => [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
[1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,1,0,1,0,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [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
[1,1,0,0,1,1,0,1,0,0]
 => [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
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [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
[1,1,0,1,1,0,0,1,0,0]
 => [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
[1,1,0,1,1,0,1,0,0,0]
 => [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
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(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)
 => ([(0,1),(0,2),(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)
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => ([(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)
 => ([(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
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,2,1] => ([(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)
 => ([(0,6),(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
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,2,1] => ([(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)
 => ([(0,6),(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
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1] => ([(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)
 => ([(0,5),(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
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,2,1,1] => ([(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)
 => ([(0,5),(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
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,2,1,1] => ([(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)
 => ([(0,5),(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
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1] => ([(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)
 => ([(0,4),(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
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,2,1,2] => ([(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)
 => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,2,2,1] => ([(0,6),(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),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,2,2,1] => ([(0,6),(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),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,2,1,1,1] => ([(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)
 => ([(0,4),(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
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,2,1,2] => ([(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)
 => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,2,1,1,1] => ([(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)
 => ([(0,4),(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
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,2,1,1,1] => ([(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)
 => ([(0,4),(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
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,2,1,2] => ([(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)
 => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,2,2,1] => ([(0,6),(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),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,2,2,1] => ([(0,6),(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),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,2,2,1] => ([(0,6),(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),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [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
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [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
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [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
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [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
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [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
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [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
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [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
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [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
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [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)
 => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [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
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [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
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [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
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [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
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [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
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [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
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [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
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [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
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [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
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [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
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [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
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [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
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [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
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [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
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [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
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [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
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [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
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [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
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [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
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [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

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

$D$ .

Matching statistic: St000679
(load all 2 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [[[]]]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[[]],[[]]]]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[[],[[]]]]]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[[]]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[[],[],[],[[]]]]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[[],[],[]],[[]]]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [[],[[],[],[[]]]]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[[],[],[[[]]]]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [[[],[]],[],[[]]]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[[],[[]],[[]]]]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[],[[],[]],[[]]]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [[],[],[[],[[]]]]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[[],[[],[[]]]]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [[[[],[]]],[[]]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[],[[[]],[[]]]]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[],[[[],[[]]]]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[[],[[[[]]]]]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[]],[],[],[[]]]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[[[]],[],[[]]]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[[[]],[]],[[]]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[[]],[[],[[]]]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[[[]],[[[]]]]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[],[[]],[],[[]]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[[[],[]],[[]]]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[],[],[[]],[[]]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[[[]]]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [[[[],[],[[]]]]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[[],[[]]],[[]]]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [[[],[]],[[[]]]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[],[[],[[[]]]]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [[[[],[[[]]]]]]
 => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [[[],[],[],[],[],[[]]]]
 => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [[[],[],[],[],[]],[[]]]
 => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [[[],[],[],[],[[[]]]]]
 => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [[[],[],[],[]],[],[[]]]
 => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [[[],[],[],[[]],[[]]]]
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [[[],[],[],[[],[[]]]]]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
 => [[[[],[],[],[]]],[[]]]
 => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [[[],[],[],[[[[]]]]]]
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [[[],[],[]],[],[],[[]]]
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [[[],[],[[]],[],[[]]]]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [[[],[],[[]],[]],[[]]]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [[[],[],[[]],[[[]]]]]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [[[],[],[[],[]],[[]]]]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [[[],[],[[],[],[[]]]]]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [[[],[],[[],[]]],[[]]]
 => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [[[],[],[[],[[[]]]]]]
 => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
 => [[[[],[],[]]],[],[[]]]
 => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [[[],[],[[[]]],[[]]]]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [[[],[],[[[]],[[]]]]]
 => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [[[],[],[[[],[[]]]]]]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
 => [[[[[],[],[]]]],[[]]]
 => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [[[],[],[[[[[]]]]]]]
 => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [[[],[[]],[],[],[[]]]]
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [[[],[[]],[],[]],[[]]]
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [[[],[[]],[],[[[]]]]]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [[[],[[]],[]],[],[[]]]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [[[],[[]],[[]],[[]]]]
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [[[],[[]],[[],[[]]]]]
 => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
 => [[[[],[[]],[]]],[[]]]
 => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [[[],[[]],[[[[]]]]]]
 => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [[[],[[],[]],[],[[]]]]
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => [[[],[[],[]],[]],[[]]]
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [[[],[[],[]],[[[]]]]]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [[[],[[],[],[]],[[]]]]
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [[[],[[],[],[],[[]]]]]
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => [[[],[[],[],[]]],[[]]]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [[[],[[],[],[[[]]]]]]
 => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [[[],[[],[]]],[],[[]]]
 => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [[[],[[],[[]]],[[]]]]
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [[[],[[],[[]],[[]]]]]
 => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [[[],[[],[[],[[]]]]]]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
 => [[[[],[[],[]]]],[[]]]
 => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [[[],[[],[[[[]]]]]]]
 => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[],[]]],[],[],[[]]]
 => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [[[],[[[]]],[],[[]]]]
 => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [[[],[[[]]],[]],[[]]]
 => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [[[],[[[]]],[[[]]]]]
 => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [[[],[[[]],[]],[[]]]]
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [[[],[[[]],[],[[]]]]]
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [[[],[[[]],[]]],[[]]]
 => ? = 2

Description

The pruning number of an ordered tree. A hanging branch of an ordered tree is a proper factor of the form

$[^r]^r$ for some

$r\geq 1$ . A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch. A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to

$[]$ .

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => [.,.]
 => ([],1)
 => ([],1)
 => ? = 1 - 2
[1,0,1,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 - 2
[1,1,0,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 - 2
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 2
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 0 = 2 - 2
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 2
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 2
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 2
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[[[[.,.],.],.],[.,.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[[[.,.],.],[[.,.],.]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[[[.,.],[[.,.],.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[[.,.],[[[.,.],.],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[[.,[.,.]],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,[.,.]],.],.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [[[.,.],[.,[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(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.

Matching statistic: St001359
(load all 3 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => [.,.]
 => [1] => 1
[1,0,1,0]
 => [1,2] => [.,[.,.]]
 => [2,1] => 1
[1,1,0,0]
 => [2,1] => [[.,.],.]
 => [1,2] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,3,6,7,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,3,7,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,3,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,3,4,5,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,3,4,7,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,3,7,4,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [2,7,6,5,4,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [2,6,7,5,4,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,5,6] => [.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [2,6,5,7,4,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [2,4,7,6,5,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => [2,4,6,7,5,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,6,4,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [2,4,7,6,5,3,1] => ? = 2

Description

The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. In other words, this is

$2^k$ where

$k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition. The generating function for the number of equivalence classes,

$f(n)$ , is

$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$

Matching statistic: St000486
(load all 3 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => [.,.]
 => [1] => ? = 1 - 1
[1,0,1,0]
 => [1,2] => [.,[.,.]]
 => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [[.,.],.]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,3,6,7,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,3,7,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,3,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,3,4,5,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,3,4,7,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,3,7,4,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [2,7,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [2,6,7,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,5,6] => [.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [2,6,5,7,4,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [2,4,7,6,5,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => [2,4,6,7,5,3,1] => ? = 2 - 1

Description

The number of cycles of length at least 3 of a permutation.

Matching statistic: St001085

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [3,1,2] => [2,3,1] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [4,1,3,2] => [2,4,3,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,1,2,4] => [3,4,1,5,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [3,4,1,2,5] => [3,4,1,2,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,3,1,2,5] => [3,4,2,1,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,4,5,1,3] => [4,1,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,1,3] => [4,2,5,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [2,3,5,1,4] => [4,1,2,5,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [2,3,4,1,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,2,4,1,5] => [4,2,1,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,5,3,1,4] => [4,1,3,5,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,1,5] => [4,1,3,2,5] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [4,2,3,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,3,5,2] => [2,5,3,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,5,1,4,2] => [3,5,1,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,4,1,5,2] => [3,5,1,2,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,3,1,5,2] => [3,5,2,1,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,2,5,3] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [5,1,2,3,4] => [2,3,4,5,1] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [5,3,1,2,4] => [3,4,2,5,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,5,4,1,3] => [4,1,5,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [5,2,4,1,3] => [4,2,5,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [5,2,3,1,4] => [4,2,3,5,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [5,3,2,1,4] => [4,3,2,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,3,2,1] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [6,4,7,1,2,3,5] => [4,5,6,2,7,1,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [5,4,6,1,2,3,7] => [4,5,6,2,1,3,7] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [4,7,5,1,2,3,6] => [4,5,6,1,3,7,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [4,6,5,1,2,3,7] => [4,5,6,1,3,2,7] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [6,4,5,1,2,3,7] => [4,5,6,2,3,1,7] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [6,3,5,7,1,2,4] => [5,6,2,7,3,1,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [5,7,3,6,1,2,4] => [5,6,3,7,1,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [5,6,3,7,1,2,4] => [5,6,3,7,1,2,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [6,5,3,7,1,2,4] => [5,6,3,7,2,1,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [6,3,4,7,1,2,5] => [5,6,2,3,7,1,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [5,3,4,6,1,2,7] => [5,6,2,3,1,4,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [7,4,5,3,6,2,1] => [4,7,3,5,1,2,6] => [5,6,3,1,4,7,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [4,6,3,5,1,2,7] => [5,6,3,1,4,2,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [4,5,3,6,1,2,7] => [5,6,3,1,2,4,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [5,4,3,6,1,2,7] => [5,6,3,2,1,4,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [3,6,7,4,1,2,5] => [5,6,1,4,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [6,3,7,4,1,2,5] => [5,6,2,4,7,1,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [7,5,3,4,6,2,1] => [3,5,7,4,1,2,6] => [5,6,1,4,2,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [3,5,6,4,1,2,7] => [5,6,1,4,2,3,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [5,3,6,4,1,2,7] => [5,6,2,4,1,3,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,4,3,5,6,2,1] => [3,4,7,5,1,2,6] => [5,6,1,2,4,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [3,4,6,5,1,2,7] => [5,6,1,2,4,3,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [6,3,4,5,1,2,7] => [5,6,2,3,4,1,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [6,4,3,5,1,2,7] => [5,6,3,2,4,1,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [3,7,5,4,1,2,6] => [5,6,1,4,3,7,2] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [3,6,5,4,1,2,7] => [5,6,1,4,3,2,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [6,3,5,4,1,2,7] => [5,6,2,4,3,1,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [6,5,3,4,1,2,7] => [5,6,3,4,2,1,7] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => [6,1,7,2,3,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [6,2,4,5,7,1,3] => [6,2,7,3,4,1,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [5,7,2,4,6,1,3] => [6,3,7,4,1,5,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [5,6,2,4,7,1,3] => [6,3,7,4,1,2,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [6,5,2,4,7,1,3] => [6,3,7,4,2,1,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [4,6,7,2,5,1,3] => [6,4,7,1,5,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,2,3,1] => [4,5,7,2,6,1,3] => [6,4,7,1,2,5,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [4,5,6,2,7,1,3] => [6,4,7,1,2,3,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [5,4,6,2,7,1,3] => [6,4,7,2,1,3,5] => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [4,7,5,2,6,1,3] => [6,4,7,1,3,5,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,2,3,1] => [4,6,5,2,7,1,3] => [6,4,7,1,3,2,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [6,4,5,2,7,1,3] => [6,4,7,2,3,1,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [6,5,4,2,7,1,3] => [6,4,7,3,2,1,5] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,4,1] => [6,2,3,5,7,1,4] => [6,2,3,7,4,1,5] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,2,4,1] => [5,7,2,3,6,1,4] => [6,3,4,7,1,5,2] => ? = 2 - 1

Description

The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern

$213$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.

Matching statistic: St001335
(load all 2 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,0]
 => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [8,3,4,5,6,7,1,2] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2,8,4,5,6,7,1,3] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [7,3,4,5,6,1,8,2] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [2,3,8,5,6,7,1,4] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [8,3,7,5,6,1,2,4] => ([(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,7,4,5,6,1,8,3] => ([(0,7),(1,6),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [6,3,4,5,1,7,8,2] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [8,6,4,5,1,7,2,3] => ([(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [2,8,7,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [8,3,6,5,1,7,2,4] => ([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [7,6,4,5,1,2,8,3] => ([(0,5),(1,3),(1,4),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,3,4,8,6,7,1,5] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [8,3,4,7,6,1,2,5] => ([(0,3),(0,4),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [2,8,4,7,6,1,3,5] => ([(0,6),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [8,3,4,6,1,7,2,5] => ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [7,8,4,6,1,2,3,5] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [2,3,7,5,6,1,8,4] => ([(0,7),(1,7),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [7,3,6,5,1,2,8,4] => ([(0,4),(1,2),(1,3),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,6,4,5,1,7,8,3] => ([(0,7),(1,7),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [5,3,4,1,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [8,5,4,1,6,7,2,3] => ([(0,3),(0,4),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [2,8,6,5,1,7,3,4] => ([(0,4),(1,2),(1,3),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [8,3,5,1,6,7,2,4] => ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [7,5,4,1,6,2,8,3] => ([(0,6),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [7,8,5,1,6,2,3,4] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [2,3,8,7,6,1,4,5] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [7,3,8,6,1,2,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [2,8,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [8,3,4,1,6,7,2,5] => ([(0,4),(0,5),(0,7),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [7,8,4,1,6,2,3,5] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [2,7,6,5,1,3,8,4] => ([(0,4),(1,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [7,3,5,1,6,2,8,4] => ([(0,6),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [6,5,4,1,2,7,8,3] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [6,8,5,1,2,7,3,4] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,7,8,6,1,3,4,5] => ([(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [7,3,8,1,6,2,4,5] => ([(0,3),(0,5),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [6,8,4,1,2,7,3,5] => ([(0,3),(0,5),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [6,7,5,1,2,3,8,4] => ([(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [8,3,4,5,7,1,2,6] => ([(0,4),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [2,8,4,5,7,1,3,6] => ([(0,6),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [8,3,4,5,1,7,2,6] => ([(0,4),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [7,8,4,5,1,2,3,6] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [2,3,8,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [7,3,8,5,1,2,4,6] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [2,8,4,5,1,7,3,6] => ([(0,6),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [5,3,4,1,8,7,2,6] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [8,5,4,1,7,2,3,6] => ([(0,4),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [2,7,8,5,1,3,4,6] => ([(0,4),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 - 1

Description

The cardinality of a minimal cycle-isolating set of a graph. Let

$\mathcal F$ be a set of graphs. A set of vertices

$S$ is

$\mathcal F$ -isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of

$S$ does not contain any graph in

$\mathcal F$ . This statistic returns the cardinality of the smallest isolating set when

$\mathcal F$ contains all cycles.

The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000360The number of occurrences of the pattern 32-1. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000397The Strahler number of a rooted tree. St000298The order dimension or Dushnik-Miller dimension of a poset. St000537The cutwidth of a graph. St000778The metric dimension of a graph. St000846The maximal number of elements covering an element of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001270The bandwidth of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St001638The book thickness of a graph. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000996The number of exclusive left-to-right maxima of a permutation. St000307The number of rowmotion orbits of a poset. St000862The number of parts of the shifted shape of a permutation. St001330The hat guessing number of a graph. St000897The number of different multiplicities of parts of an integer partition. St000779The tier of a permutation. St000640The rank of the largest boolean interval in a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001728The number of invisible descents of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000805The number of peaks of the associated bargraph. St001729The number of visible descents of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000354The number of recoils of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000711The number of big exceedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001469The holeyness of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001665The number of pure excedances of a permutation. St000023The number of inner peaks of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000768The number of peaks in an integer composition. St001960The number of descents of a permutation minus one if its first entry is not one. St000454The largest eigenvalue of a graph if it is integral.