- FindStat

Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000679
(load all 3 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000396
(load all 5 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree

$[[[.,.],[.,.]],[[.,.],[.,.]]]$ : its register function is 3, whereas the dimension of the associated poset is 2.

Matching statistic: St000920

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [[]]
 => [.,.]
 => [1,0]
 => 1
[.,[.,.]]
 => [[[]]]
 => [[.,.],.]
 => [1,1,0,0]
 => 1
[[.,.],.]
 => [[],[]]
 => [.,[.,.]]
 => [1,0,1,0]
 => 1
[.,[.,[.,.]]]
 => [[[[]]]]
 => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => 1
[.,[[.,.],.]]
 => [[[],[]]]
 => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2
[[.,.],[.,.]]
 => [[],[[]]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [[[]],[]]
 => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [[],[],[]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1
[.,[.,[.,[.,.]]]]
 => [[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2
[.,[[.,.],[.,.]]]
 => [[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2
[.,[[.,[.,.]],.]]
 => [[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => 2
[.,[[[.,.],.],.]]
 => [[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 2
[[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[.,.],[[.,.],.]]
 => [[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2
[[.,[.,.]],[.,.]]
 => [[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => [[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 2
[[[.,.],[.,.]],.]
 => [[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[.,[.,.]],.],.]
 => [[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[[.,.],.],.],.]
 => [[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [[[[[[]]]]]]
 => [[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [[[[[],[]]]]]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[.,[.,[[.,.],[.,.]]]]
 => [[[[],[[]]]]]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [[[[[]],[]]]]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [[[[],[],[]]]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [[[],[[[]]]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [[[],[[],[]]]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [[[[]],[[]]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [[[],[],[[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [[[[[]]],[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [[[[],[]],[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [[[],[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [[[[]],[],[]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[.,[[[[.,.],.],.],.]]
 => [[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[.,.],[[.,.],[.,.]]]
 => [[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[.,.],[[[.,.],.],.]]
 => [[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[[.,[.,.]],.],[.,.]]
 => [[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[[.,.],.],.],[.,.]]
 => [[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [[[]],[[[[],[],[]]]]]
 => [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => ? = 2
[[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
 => [[[]],[[[],[[],[]]]]]
 => [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [[[]],[[[],[],[[]]]]]
 => [[.,.],[[[.,.],[.,[[.,.],.]]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 2
[[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]
 => [[[]],[[[[[]]],[]]]]
 => [[.,.],[[[[[.,.],.],.],[.,.]],.]]
 => [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [[[]],[[[],[[]],[]]]]
 => [[.,.],[[[.,.],[[.,[.,.]],.]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 2
[[.,[.,.]],[.,[[[.,[.,.]],.],.]]]
 => [[[]],[[[[]],[],[]]]]
 => [[.,.],[[[[.,.],.],[.,[.,.]]],.]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
 => [[[]],[[[],[],[],[]]]]
 => [[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 2
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
 => [[[]],[[],[[[],[]]]]]
 => [[.,.],[[.,[[[.,.],[.,.]],.]],.]]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2
[[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
 => [[[]],[[],[[],[[]]]]]
 => [[.,.],[[.,[[.,.],[[.,.],.]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 2
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
 => [[[]],[[],[[],[],[]]]]
 => [[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[[.,[.,.]],[[[.,.],.],[.,[.,.]]]]
 => [[[]],[[],[],[[[]]]]]
 => [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 2
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [[[]],[[],[],[[],[]]]]
 => [[.,.],[[.,[.,[[.,.],[.,.]]]],.]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
 => [[[]],[[[[]]],[[]]]]
 => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 2
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
 => [[[]],[[],[[]],[[]]]]
 => [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 2
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
 => [[[]],[[[]],[],[[]]]]
 => [[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [[[]],[[],[],[],[[]]]]
 => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 2
[[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
 => [[[]],[[[[[]]]],[]]]
 => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [[[]],[[[[],[]]],[]]]
 => [[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [[[]],[[[[]],[]],[]]]
 => [[.,.],[[[[.,.],.],[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [[[]],[[[],[],[]],[]]]
 => [[.,.],[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[[.,[.,.]],[[[.,.],[.,[.,.]]],.]]
 => [[[]],[[],[[[]]],[]]]
 => [[.,.],[[.,.],[[[.,[.,.]],.],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
 => [[[]],[[],[[],[]],[]]]
 => [[.,.],[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [[[]],[[[]],[[]],[]]]
 => [[.,.],[[[.,.],.],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 2
[[.,[.,.]],[[[[.,.],.],[.,.]],.]]
 => [[[]],[[],[],[[]],[]]]
 => [[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [[[]],[[[[]]],[],[]]]
 => [[.,.],[[[[.,.],.],.],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 2
[[.,[.,.]],[[[[.,.],[.,.]],.],.]]
 => [[[]],[[],[[]],[],[]]]
 => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 2
[[.,[.,.]],[[[[.,[.,.]],.],.],.]]
 => [[[]],[[[]],[],[],[]]]
 => [[.,.],[[[.,.],.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
 => [[[]],[[],[],[],[],[]]]
 => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 2
[[[.,[.,.]],.],[.,[.,[[.,.],.]]]]
 => [[[]],[],[[[[],[]]]]]
 => [[.,.],[.,[[[[.,.],[.,.]],.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 2
[[[.,[.,.]],.],[.,[[.,.],[.,.]]]]
 => [[[]],[],[[[],[[]]]]]
 => [[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 2
[[[.,[.,.]],.],[.,[[.,[.,.]],.]]]
 => [[[]],[],[[[[]],[]]]]
 => [[.,.],[.,[[[[.,.],.],[.,.]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 2
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
 => [[[]],[],[[[],[],[]]]]
 => [[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 2
[[[.,[.,.]],.],[[.,.],[.,[.,.]]]]
 => [[[]],[],[[],[[[]]]]]
 => [[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[[[.,[.,.]],.],[[.,.],[[.,.],.]]]
 => [[[]],[],[[],[[],[]]]]
 => [[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[[[.,[.,.]],.],[[.,[.,.]],[.,.]]]
 => [[[]],[],[[[]],[[]]]]
 => [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [[[]],[],[[],[],[[]]]]
 => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[[[.,[.,.]],.],[[[.,.],[.,.]],.]]
 => [[[]],[],[[],[[]],[]]]
 => [[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
 => [[[]],[],[[],[],[],[]]]
 => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[[[.,[.,.]],[.,.]],[[[.,.],.],.]]
 => [[[]],[[]],[[],[],[]]]
 => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 2
[[[[.,[.,.]],.],.],[.,[[.,.],.]]]
 => [[[]],[],[],[[[],[]]]]
 => [[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
 => [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [[[]],[],[],[[],[[]]]]
 => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 2
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
 => [[[]],[],[],[[[]],[]]]
 => [[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
[[[[.,[.,.]],.],.],[[[.,.],.],.]]
 => [[[]],[],[],[[],[],[]]]
 => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[[[.,[.,.]],[[.,.],.]],[.,[.,.]]]
 => [[[]],[[],[]],[[[]]]]
 => [[.,.],[[.,[.,.]],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[[[.,[.,.]],[[.,.],.]],[[.,.],.]]
 => [[[]],[[],[]],[[],[]]]
 => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
 => [[[]],[],[[]],[[],[]]]
 => [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 2
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
 => [[[]],[[]],[],[[],[]]]
 => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[[[[.,[.,[.,.]]],.],.],[[.,.],.]]
 => [[[[]]],[],[],[[],[]]]
 => [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [[[],[]],[],[],[[],[]]]
 => [[.,[.,.]],[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[[[[.,[.,.]],.],.],.],[[.,.],.]]
 => [[[]],[],[],[],[[],[]]]
 => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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: St000862
(load all 2 compositions to match this statistic)

Mapping

Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of parts of the shifted shape.

Matching statistic: St001741

Mapping

Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest integer such that all patterns of this size are contained in the permutation.

Matching statistic: St001085

Mapping

Mp00014: Binary trees —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] => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [2,3,1] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,1,3,2] => [2,4,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,5,1,2,4] => [3,4,1,5,2] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [3,4,1,2,5] => [3,4,1,2,5] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,3,1,2,5] => [3,4,2,1,5] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [2,4,5,1,3] => [4,1,5,2,3] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [4,2,5,1,3] => [4,2,5,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [2,3,5,1,4] => [4,1,2,5,3] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,5,3,1,4] => [4,1,3,5,2] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [2,3,4,1,5] => [4,1,2,3,5] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,2,4,1,5] => [4,2,1,3,5] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,4,3,1,5] => [4,1,3,2,5] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,2,3,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,1,3,5,2] => [2,5,3,1,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,5,1,4,2] => [3,5,1,4,2] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,4,1,5,2] => [3,5,1,2,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,3,1,5,2] => [3,5,2,1,4] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,1,2,5,3] => [2,3,5,1,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,4,5,3,2] => [1,5,4,2,3] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,1,5,3,2] => [2,5,4,1,3] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,5,4,1,3] => [4,1,5,3,2] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,3,5,4,2] => [1,5,2,4,3] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [7,5,6,4,3,2,1] => [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2 - 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
 => [5,6,7,4,3,2,1] => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 2 - 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [7,6,4,5,3,2,1] => [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 2 - 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [6,7,4,5,3,2,1] => [6,4,7,1,2,3,5] => [4,5,6,2,7,1,3] => ? = 2 - 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [7,5,4,6,3,2,1] => [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 2 - 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [7,4,5,6,3,2,1] => [4,7,5,1,2,3,6] => [4,5,6,1,3,7,2] => ? = 2 - 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [5,6,4,7,3,2,1] => [5,4,6,1,2,3,7] => [4,5,6,2,1,3,7] => ? = 2 - 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [6,4,5,7,3,2,1] => [4,6,5,1,2,3,7] => [4,5,6,1,3,2,7] => ? = 2 - 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [5,4,6,7,3,2,1] => [6,4,5,1,2,3,7] => [4,5,6,2,3,1,7] => ? = 2 - 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => [5,6,1,7,2,3,4] => ? = 2 - 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [6,7,5,3,4,2,1] => [6,3,5,7,1,2,4] => [5,6,2,7,3,1,4] => ? = 2 - 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [7,5,6,3,4,2,1] => [5,7,3,6,1,2,4] => [5,6,3,7,1,4,2] => ? = 2 - 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [6,5,7,3,4,2,1] => [5,6,3,7,1,2,4] => [5,6,3,7,1,2,4] => ? = 2 - 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [5,6,7,3,4,2,1] => [6,5,3,7,1,2,4] => [5,6,3,7,2,1,4] => ? = 2 - 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [7,6,4,3,5,2,1] => [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 2 - 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [6,7,4,3,5,2,1] => [6,3,4,7,1,2,5] => [5,6,2,3,7,1,4] => ? = 2 - 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [7,6,3,4,5,2,1] => [3,6,7,4,1,2,5] => [5,6,1,4,7,2,3] => ? = 2 - 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => [6,7,3,4,5,2,1] => [6,3,7,4,1,2,5] => [5,6,2,4,7,1,3] => ? = 2 - 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [7,4,5,3,6,2,1] => [4,7,3,5,1,2,6] => [5,6,3,1,4,7,2] => ? = 2 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [7,5,3,4,6,2,1] => [3,5,7,4,1,2,6] => [5,6,1,4,2,7,3] => ? = 2 - 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [7,4,3,5,6,2,1] => [3,4,7,5,1,2,6] => [5,6,1,2,4,7,3] => ? = 2 - 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [7,3,4,5,6,2,1] => [3,7,5,4,1,2,6] => [5,6,1,4,3,7,2] => ? = 2 - 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [5,6,4,3,7,2,1] => [5,3,4,6,1,2,7] => [5,6,2,3,1,4,7] => ? = 2 - 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [6,4,5,3,7,2,1] => [4,6,3,5,1,2,7] => [5,6,3,1,4,2,7] => ? = 2 - 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [5,4,6,3,7,2,1] => [4,5,3,6,1,2,7] => [5,6,3,1,2,4,7] => ? = 2 - 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => [5,4,3,6,1,2,7] => [5,6,3,2,1,4,7] => ? = 2 - 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [6,5,3,4,7,2,1] => [3,5,6,4,1,2,7] => [5,6,1,4,2,3,7] => ? = 2 - 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => [5,6,3,4,7,2,1] => [5,3,6,4,1,2,7] => [5,6,2,4,1,3,7] => ? = 2 - 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [6,4,3,5,7,2,1] => [3,4,6,5,1,2,7] => [5,6,1,2,4,3,7] => ? = 2 - 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [6,3,4,5,7,2,1] => [3,6,5,4,1,2,7] => [5,6,1,4,3,2,7] => ? = 2 - 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [5,4,3,6,7,2,1] => [6,3,4,5,1,2,7] => [5,6,2,3,4,1,7] => ? = 2 - 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [4,5,3,6,7,2,1] => [6,4,3,5,1,2,7] => [5,6,3,2,4,1,7] => ? = 2 - 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [5,3,4,6,7,2,1] => [6,3,5,4,1,2,7] => [5,6,2,4,3,1,7] => ? = 2 - 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [4,3,5,6,7,2,1] => [6,5,3,4,1,2,7] => [5,6,3,4,2,1,7] => ? = 2 - 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => [6,1,7,2,3,4,5] => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [6,7,5,4,2,3,1] => [6,2,4,5,7,1,3] => [6,2,7,3,4,1,5] => ? = 2 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [7,5,6,4,2,3,1] => [5,7,2,4,6,1,3] => [6,3,7,4,1,5,2] => ? = 2 - 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [6,5,7,4,2,3,1] => [5,6,2,4,7,1,3] => [6,3,7,4,1,2,5] => ? = 2 - 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
 => [5,6,7,4,2,3,1] => [6,5,2,4,7,1,3] => [6,3,7,4,2,1,5] => ? = 2 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [7,6,4,5,2,3,1] => [4,6,7,2,5,1,3] => [6,4,7,1,5,2,3] => ? = 2 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [7,5,4,6,2,3,1] => [4,5,7,2,6,1,3] => [6,4,7,1,2,5,3] => ? = 2 - 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [7,4,5,6,2,3,1] => [4,7,5,2,6,1,3] => [6,4,7,1,3,5,2] => ? = 2 - 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
 => [6,5,4,7,2,3,1] => [4,5,6,2,7,1,3] => [6,4,7,1,2,3,5] => ? = 2 - 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
 => [5,6,4,7,2,3,1] => [5,4,6,2,7,1,3] => [6,4,7,2,1,3,5] => ? = 2 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [6,4,5,7,2,3,1] => [4,6,5,2,7,1,3] => [6,4,7,1,3,2,5] => ? = 2 - 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
 => [5,4,6,7,2,3,1] => [6,4,5,2,7,1,3] => [6,4,7,2,3,1,5] => ? = 2 - 1
[.,[[.,.],[[[[.,.],.],.],.]]]
 => [4,5,6,7,2,3,1] => [6,5,4,2,7,1,3] => [6,4,7,3,2,1,5] => ? = 2 - 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [6,7,5,3,2,4,1] => [6,2,3,5,7,1,4] => [6,2,3,7,4,1,5] => ? = 2 - 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
 => [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

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ 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]
 => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[.,[.,.]]
 => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[.,.],.]
 => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [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
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [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
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [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
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [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
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [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
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [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
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [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
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [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
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [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
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [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
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [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
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [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
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [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
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [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
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [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
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [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
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [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]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [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
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [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
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [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
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [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
[.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [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
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [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
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [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
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [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
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [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
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [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
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [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
[.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [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
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [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
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [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
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [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
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [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
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [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
[.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [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
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [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
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [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
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [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
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [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
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [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
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [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
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [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
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [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
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [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
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [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
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [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
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [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
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [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
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [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
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [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
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [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
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [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
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [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
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [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
[.,[.,[[[[[.,.],.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [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
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [4,3,1,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [8,4,1,5,6,7,2,3] => ([(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
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [7,4,1,5,6,2,8,3] => ([(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
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [8,3,1,5,6,7,2,4] => ([(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
[.,[[.,.],[.,[[[.,.],.],.]]]]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [7,8,1,5,6,2,3,4] => ([(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
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [6,4,1,5,2,7,8,3] => ([(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
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [6,8,1,5,2,7,3,4] => ([(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
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [7,3,1,5,6,2,8,4] => ([(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
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [6,7,1,5,2,3,8,4] => ([(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
[.,[[.,.],[[.,[.,[.,.]]],.]]]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [4,3,1,8,6,7,2,5] => ([(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
[.,[[.,.],[[.,[[.,.],.]],.]]]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [8,4,1,7,6,2,3,5] => ([(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)
 => ? = 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.

Matching statistic: St001235

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%

Values

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

Description

The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Matching statistic: St000397
(load all 2 compositions to match this statistic)

Mapping

Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%

Values

[.,.]
 => [.,.]
 => [.,.]
 => [[],[]]
 => 2 = 1 + 1
[.,[.,.]]
 => [[.,.],.]
 => [[.,.],.]
 => [[[],[]],[]]
 => 2 = 1 + 1
[[.,.],.]
 => [.,[.,.]]
 => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 1 + 1
[.,[.,[.,.]]]
 => [[[.,.],.],.]
 => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 3 = 2 + 1
[[.,.],[.,.]]
 => [[.,.],[.,.]]
 => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [.,[[.,.],.]]
 => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 2 = 1 + 1
[.,[.,[[.,.],.]]]
 => [[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 3 = 2 + 1
[.,[[.,.],[.,.]]]
 => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 3 = 2 + 1
[.,[[.,[.,.]],.]]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 3 = 2 + 1
[.,[[[.,.],.],.]]
 => [[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 3 = 2 + 1
[[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 3 = 2 + 1
[[.,[.,.]],[.,.]]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 3 = 2 + 1
[[[.,.],[.,.]],.]
 => [.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [[[[[.,.],.],.],.],.]
 => [[[[[.,.],.],.],.],.]
 => [[[[[[],[]],[]],[]],[]],[]]
 => 2 = 1 + 1
[.,[.,[.,[[.,.],.]]]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[.,.],.],.],[.,.]]
 => [[[[[],[]],[]],[]],[[],[]]]
 => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
 => [[[[.,.],[.,.]],.],.]
 => [[[[.,.],.],[.,.]],.]
 => [[[[[],[]],[]],[[],[]]],[]]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [[[.,[[.,.],.]],.],.]
 => [[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
 => [[[[.,.],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],.]
 => [[[[[],[]],[[],[]]],[]],[]]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[.,.],[.,.]],[.,.]]
 => [[[[],[]],[[],[]]],[[],[]]]
 => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
 => [[[.,.],[[.,.],.]],.]
 => [[[.,.],[[.,.],.]],.]
 => [[[[],[]],[[[],[]],[]]],[]]
 => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
 => [[[.,.],[.,[.,.]]],.]
 => [[[.,.],[.,[.,.]]],.]
 => [[[[],[]],[[],[[],[]]]],[]]
 => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [[.,[[.,[.,.]],.]],.]
 => [[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
 => [[.,[.,[[.,.],.]]],.]
 => [[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[[],[]]],[]],[]],[]]
 => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[[],[]]],[[],[]]],[]]
 => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
 => [[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[[],[]],[]]],[]],[]]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[[],[[],[]]]],[]],[]]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[[],[]],[]],[]]],[]]
 => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[[],[]],[[],[]]]],[]]
 => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[[],[[],[]]],[]]],[]]
 => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[[],[[[],[]],[]]]],[]]
 => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[[],[[],[[],[]]]]],[]]
 => 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => [[[[[[[[],[]],[]],[]],[]],[]],[]],[]]
 => ? = 1 + 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [[[[[[[],[]],[]],[]],[]],[]],[[],[]]]
 => ? = 2 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [[[[[[.,.],.],.],.],[.,.]],.]
 => [[[[[[[],[]],[]],[]],[]],[[],[]]],[]]
 => ? = 2 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [[[[[.,[[.,.],.]],.],.],.],.]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => [[[[[[],[]],[]],[]],[]],[[[],[]],[]]]
 => ? = 2 + 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
 => [[[[[.,[.,[.,.]]],.],.],.],.]
 => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [[[[[[],[]],[]],[]],[]],[[],[[],[]]]]
 => ? = 2 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [[[[[[.,.],.],[.,.]],.],.],.]
 => [[[[[[.,.],.],.],[.,.]],.],.]
 => [[[[[[[],[]],[]],[]],[[],[]]],[]],[]]
 => ? = 2 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [[[[[.,[.,.]],[.,.]],.],.],.]
 => [[[[[.,.],.],.],[.,.]],[.,.]]
 => [[[[[[],[]],[]],[]],[[],[]]],[[],[]]]
 => ? = 2 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [[[[[.,.],[[.,.],.]],.],.],.]
 => [[[[[.,.],.],.],[[.,.],.]],.]
 => [[[[[[],[]],[]],[]],[[[],[]],[]]],[]]
 => ? = 2 + 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [[[[[.,.],[.,[.,.]]],.],.],.]
 => [[[[[.,.],.],.],[.,[.,.]]],.]
 => [[[[[[],[]],[]],[]],[[],[[],[]]]],[]]
 => ? = 2 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [[[[.,[[[.,.],.],.]],.],.],.]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => [[[[[],[]],[]],[]],[[[[],[]],[]],[]]]
 => ? = 2 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [[[[.,[[.,[.,.]],.]],.],.],.]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [[[[[],[]],[]],[]],[[[],[]],[[],[]]]]
 => ? = 2 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [[[[.,[[.,.],[.,.]]],.],.],.]
 => [[[[.,.],.],.],[[.,[.,.]],.]]
 => [[[[[],[]],[]],[]],[[[],[[],[]]],[]]]
 => ? = 2 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [[[[.,[.,[[.,.],.]]],.],.],.]
 => [[[[.,.],.],.],[.,[[.,.],.]]]
 => [[[[[],[]],[]],[]],[[],[[[],[]],[]]]]
 => ? = 2 + 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
 => [[[[.,[.,[.,[.,.]]]],.],.],.]
 => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [[[[[],[]],[]],[]],[[],[[],[[],[]]]]]
 => ? = 2 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [[[[[[.,.],.],.],[.,.]],.],.]
 => [[[[[[.,.],.],[.,.]],.],.],.]
 => [[[[[[[],[]],[]],[[],[]]],[]],[]],[]]
 => ? = 2 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [[[[[.,[.,.]],.],[.,.]],.],.]
 => [[[[[.,.],.],[.,.]],.],[.,.]]
 => [[[[[[],[]],[]],[[],[]]],[]],[[],[]]]
 => ? = 2 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [[[[[[],[]],[]],[[],[]]],[[],[]]],[]]
 => ? = 2 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [[[[.,[[.,.],.]],[.,.]],.],.]
 => [[[[.,.],.],[.,.]],[[.,.],.]]
 => [[[[[],[]],[]],[[],[]]],[[[],[]],[]]]
 => ? = 2 + 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [[[[.,[.,[.,.]]],[.,.]],.],.]
 => [[[[.,.],.],[.,.]],[.,[.,.]]]
 => [[[[[],[]],[]],[[],[]]],[[],[[],[]]]]
 => ? = 2 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [[[[[.,.],.],[[.,.],.]],.],.]
 => [[[[[.,.],.],[[.,.],.]],.],.]
 => [[[[[[],[]],[]],[[[],[]],[]]],[]],[]]
 => ? = 2 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [[[[.,[.,.]],[[.,.],.]],.],.]
 => [[[[.,.],.],[[.,.],.]],[.,.]]
 => [[[[[],[]],[]],[[[],[]],[]]],[[],[]]]
 => ? = 2 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [[[[[[],[]],[]],[[],[[],[]]]],[]],[]]
 => ? = 2 + 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => [[[[[],[]],[]],[[],[[],[]]]],[[],[]]]
 => ? = 2 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [[[[.,.],[[[.,.],.],.]],.],.]
 => [[[[.,.],.],[[[.,.],.],.]],.]
 => [[[[[],[]],[]],[[[[],[]],[]],[]]],[]]
 => ? = 2 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [[[[.,.],[[.,[.,.]],.]],.],.]
 => [[[[.,.],.],[[.,.],[.,.]]],.]
 => [[[[[],[]],[]],[[[],[]],[[],[]]]],[]]
 => ? = 2 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [[[[.,.],[[.,.],[.,.]]],.],.]
 => [[[[.,.],.],[[.,[.,.]],.]],.]
 => [[[[[],[]],[]],[[[],[[],[]]],[]]],[]]
 => ? = 2 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [[[[.,.],[.,[[.,.],.]]],.],.]
 => [[[[.,.],.],[.,[[.,.],.]]],.]
 => [[[[[],[]],[]],[[],[[[],[]],[]]]],[]]
 => ? = 2 + 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [[[[.,.],[.,[.,[.,.]]]],.],.]
 => [[[[.,.],.],[.,[.,[.,.]]]],.]
 => [[[[[],[]],[]],[[],[[],[[],[]]]]],[]]
 => ? = 2 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [[[.,[[[[.,.],.],.],.]],.],.]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => [[[[],[]],[]],[[[[[],[]],[]],[]],[]]]
 => ? = 2 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [[[.,[[[.,[.,.]],.],.]],.],.]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [[[[],[]],[]],[[[[],[]],[]],[[],[]]]]
 => ? = 2 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [[[.,[[[.,.],[.,.]],.]],.],.]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [[[[],[]],[]],[[[[],[]],[[],[]]],[]]]
 => ? = 2 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [[[.,[[.,[[.,.],.]],.]],.],.]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [[[[],[]],[]],[[[],[]],[[[],[]],[]]]]
 => ? = 2 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [[[.,[[.,[.,[.,.]]],.]],.],.]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [[[[],[]],[]],[[[],[]],[[],[[],[]]]]]
 => ? = 2 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [[[.,[[[.,.],.],[.,.]]],.],.]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [[[[],[]],[]],[[[[],[[],[]]],[]],[]]]
 => ? = 2 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => [[[.,[[.,[.,.]],[.,.]]],.],.]
 => [[[.,.],.],[[.,[.,.]],[.,.]]]
 => [[[[],[]],[]],[[[],[[],[]]],[[],[]]]]
 => ? = 2 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [[[.,[[.,.],[[.,.],.]]],.],.]
 => [[[.,.],.],[[.,[[.,.],.]],.]]
 => [[[[],[]],[]],[[[],[[[],[]],[]]],[]]]
 => ? = 2 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [[[.,[[.,.],[.,[.,.]]]],.],.]
 => [[[.,.],.],[[.,[.,[.,.]]],.]]
 => [[[[],[]],[]],[[[],[[],[[],[]]]],[]]]
 => ? = 2 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [[[.,[.,[[[.,.],.],.]]],.],.]
 => [[[.,.],.],[.,[[[.,.],.],.]]]
 => [[[[],[]],[]],[[],[[[[],[]],[]],[]]]]
 => ? = 2 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [[[.,[.,[[.,[.,.]],.]]],.],.]
 => [[[.,.],.],[.,[[.,.],[.,.]]]]
 => [[[[],[]],[]],[[],[[[],[]],[[],[]]]]]
 => ? = 2 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [[[.,[.,[[.,.],[.,.]]]],.],.]
 => [[[.,.],.],[.,[[.,[.,.]],.]]]
 => [[[[],[]],[]],[[],[[[],[[],[]]],[]]]]
 => ? = 2 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [[[.,[.,[.,[[.,.],.]]]],.],.]
 => [[[.,.],.],[.,[.,[[.,.],.]]]]
 => [[[[],[]],[]],[[],[[],[[[],[]],[]]]]]
 => ? = 2 + 1
[.,[.,[[[[[.,.],.],.],.],.]]]
 => [[[.,[.,[.,[.,[.,.]]]]],.],.]
 => [[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [[[[],[]],[]],[[],[[],[[],[[],[]]]]]]
 => ? = 2 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [[[[[[.,.],.],.],.],[.,.]],.]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [[[[[[[],[]],[[],[]]],[]],[]],[]],[]]
 => ? = 2 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [[[[[.,[.,.]],.],.],[.,.]],.]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [[[[[[],[]],[[],[]]],[]],[]],[[],[]]]
 => ? = 2 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [[[[[[],[]],[[],[]]],[]],[[],[]]],[]]
 => ? = 2 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [[[[.,[[.,.],.]],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],[[.,.],.]]
 => [[[[[],[]],[[],[]]],[]],[[[],[]],[]]]
 => ? = 2 + 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
 => [[[[.,[.,[.,.]]],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],[.,[.,.]]]
 => [[[[[],[]],[[],[]]],[]],[[],[[],[]]]]
 => ? = 2 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [[[[[[],[]],[[],[]]],[[],[]]],[]],[]]
 => ? = 2 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [[[[.,[.,.]],[.,.]],[.,.]],.]
 => [[[[.,.],[.,.]],[.,.]],[.,.]]
 => [[[[[],[]],[[],[]]],[[],[]]],[[],[]]]
 => ? = 2 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [[[[.,.],[[.,.],.]],[.,.]],.]
 => [[[[.,.],[.,.]],[[.,.],.]],.]
 => [[[[[],[]],[[],[]]],[[[],[]],[]]],[]]
 => ? = 2 + 1

Description

The Strahler number of a rooted tree.

Matching statistic: St001174
(load all 36 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%

Values

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

Description

The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ .

The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St000640The rank of the largest boolean interval in a poset. St000454The largest eigenvalue of a graph if it is integral. 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.