- 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: St000920
(load all 13 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%

Values

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

Description

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

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

Matching statistic: St000485

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%

Values

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

Description

The length of the longest cycle of a permutation.

Matching statistic: St000264
(load all 2 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St001235
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●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] => [1,3,2] => [1,2] => 2
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [3] => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2,3] => [3] => 1
[[.,[.,.]],.]
 => [2,1,3] => [1,2,3] => [3] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [3] => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,2,3] => [1,3] => 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,3,2,4] => [1,3] => 2
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => [2,2] => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,4,2] => [1,3] => 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [4] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,2,4,3] => [2,2] => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,2,3,4] => [4] => 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [1,2,3,4] => [4] => 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3,4] => [4] => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,3,2,4] => [1,3] => 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,3,4] => [4] => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,3,4] => [4] => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,2,3,4] => [4] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [4] => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => 2
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,4,2,5,3] => [1,2,2] => 2
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,3,4,2,5] => [1,4] => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,4,2,3,5] => [1,4] => 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,3,5,2,4] => [1,4] => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,2,5,3,4] => [2,3] => 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,2,4,3,5] => [2,3] => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,3,5,2,4] => [1,4] => 2
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,2,3,5,4] => [3,2] => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,4,5,2,3] => [1,4] => 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,3,2,4,5] => [1,4] => 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,2,4,5,3] => [2,3] => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,3,4,5,2] => [1,4] => 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [5] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,2,5,3,4] => [2,3] => 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,2,4,3,5] => [2,3] => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,2,3,5,4] => [3,2] => 2
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,2,4,5,3] => [2,3] => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,2,3,4,5] => [5] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [1,2,3,5,4] => [3,2] => 2
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [1,2,3,4,5] => [5] => 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,3,5,4] => [3,2] => 2
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,3,4,5] => [5] => 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [1,3,2,4,5] => [1,4] => 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [1,2,3,4,5] => [5] => 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,2,3,4,5] => [5] => 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [1,2,3,4,5] => [5] => 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4,5] => [5] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => [1,2,3,1] => ? = 3
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => [1,6,2,7,3,5,4] => [1,2,3,1] => ? = 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => [1,5,3,6,2,7,4] => [1,2,2,2] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => [1,6,2,5,3,7,4] => [1,2,2,2] => ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
 => [5,6,7,4,3,2,1] => [1,5,3,7,2,6,4] => [1,2,3,1] => ? = 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => [1,4,5,3,6,2,7] => [1,3,3] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => [1,4,5,3,7,2,6] => [1,3,3] => ? = 2
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => [1,5,3,7,2,4,6] => [1,2,4] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [4,5,7,6,3,2,1] => [1,4,6,2,5,3,7] => [1,4,2] => ? = 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => [1,6,2,5,3,4,7] => [1,2,4] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [5,6,4,7,3,2,1] => [1,5,3,4,7,2,6] => [1,3,3] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [4,6,5,7,3,2,1] => [1,4,7,2,6,3,5] => [1,3,3] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [5,4,6,7,3,2,1] => [1,5,3,6,2,4,7] => [1,2,4] => ? = 2
[.,[.,[.,[[[[.,.],.],.],.]]]]
 => [4,5,6,7,3,2,1] => [1,4,7,2,5,3,6] => [1,4,2] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => [1,3,6,2,7,4,5] => [1,3,3] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => [1,3,7,2,6,4,5] => [1,3,3] => ? = 2
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => [1,3,7,2,5,4,6] => [1,4,2] => ? = 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => [1,3,5,4,7,2,6] => [1,3,3] => ? = 2
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [3,5,6,7,4,2,1] => [1,3,6,2,5,4,7] => [1,4,2] => ? = 2
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => [1,4,6,2,3,7,5] => [1,4,2] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => [1,4,7,2,3,6,5] => [1,5,1] => ? = 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [3,4,7,6,5,2,1] => [1,3,7,2,4,6,5] => [1,5,1] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => [3,4,6,7,5,2,1] => [1,3,6,2,4,7,5] => [1,4,2] => ? = 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => [1,5,6,2,4,7,3] => [1,4,2] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [4,5,3,7,6,2,1] => [1,4,7,2,5,6,3] => [1,5,1] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [3,5,4,7,6,2,1] => [1,3,4,7,2,5,6] => [1,6] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [4,3,5,7,6,2,1] => [1,4,7,2,3,5,6] => [1,6] => ? = 2
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [3,4,5,7,6,2,1] => [1,3,5,6,2,4,7] => [1,6] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => [1,6,2,5,7,3,4] => [1,2,4] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [5,6,4,3,7,2,1] => [1,5,7,2,6,3,4] => [1,3,3] => ? = 2
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [4,6,5,3,7,2,1] => [1,4,3,5,7,2,6] => [1,2,4] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [5,4,6,3,7,2,1] => [1,5,7,2,4,3,6] => [1,4,2] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => [1,4,3,6,2,5,7] => [1,2,4] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [3,6,5,4,7,2,1] => [1,3,5,7,2,6,4] => [1,5,1] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => [3,5,6,4,7,2,1] => [1,3,6,2,5,7,4] => [1,4,2] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [4,3,6,5,7,2,1] => [1,4,5,7,2,3,6] => [1,6] => ? = 2
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [3,4,6,5,7,2,1] => [1,3,6,2,4,5,7] => [1,6] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [1,5,1] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [4,5,3,6,7,2,1] => [1,4,6,2,5,7,3] => [1,4,2] => ? = 2
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [3,5,4,6,7,2,1] => [1,3,4,6,2,5,7] => [1,6] => ? = 2
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [4,3,5,6,7,2,1] => [1,4,6,2,3,5,7] => [1,6] => ? = 2
[.,[.,[[[[[.,.],.],.],.],.]]]
 => [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,6] => ? = 2
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [2,7,6,5,4,3,1] => [1,2,7,3,6,4,5] => [2,2,3] => ? = 2
[.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [2,6,7,5,4,3,1] => [1,2,6,3,7,4,5] => [2,2,3] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => [1,2,5,4,6,3,7] => [2,2,3] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [2,6,5,7,4,3,1] => [1,2,6,3,5,4,7] => [2,3,2] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
 => [2,5,6,7,4,3,1] => [1,2,5,4,7,3,6] => [2,2,3] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [2,4,7,6,5,3,1] => [1,2,4,6,3,7,5] => [2,3,2] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [2,4,6,7,5,3,1] => [1,2,4,7,3,6,5] => [2,4,1] => ? = 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [2,5,4,7,6,3,1] => [1,2,5,6,3,4,7] => [2,5] => ? = 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

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%

Values

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

Description

The Strahler number of a rooted tree.

Matching statistic: St001335

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%

Values

[.,.]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],.]
 => [1,2] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3 - 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
 => [5,6,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [4,5,7,6,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [5,6,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [4,6,5,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [5,4,6,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
 => [4,5,6,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [3,5,6,7,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [3,4,7,6,5,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => [3,4,6,7,5,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [4,5,3,7,6,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [3,5,4,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [4,3,5,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [3,4,5,7,6,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [5,6,4,3,7,2,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [4,6,5,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [5,4,6,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [3,6,5,4,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => [3,5,6,4,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [4,3,6,5,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [3,4,6,5,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [5,4,3,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [4,5,3,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [3,5,4,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [4,3,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[.,[[[[[.,.],.],.],.],.]]]
 => [3,4,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [2,6,7,5,4,3,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [2,6,5,7,4,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
 => [2,5,6,7,4,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [2,4,7,6,5,3,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [2,4,6,7,5,3,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [2,5,4,7,6,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 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: St001174
(load all 18 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%

Values

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

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

Matching statistic: St001859
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of factors of the Stanley symmetric function associated with a permutation. For example, the Stanley symmetric function of

$\pi=321645$ equals

$20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$

Matching statistic: St000298
(load all 3 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%

Values

[.,.]
 => [1,0]
 => ([],1)
 => 1
[.,[.,.]]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 1
[[.,.],.]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 2
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 2
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 2
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 2
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 2
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 2
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 2
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 2
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 2
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 2
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 2
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 2
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 2
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 2
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 2
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 2
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 2
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 2
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 2
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 2
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 2
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 2
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[.,[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 2
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 2
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 2
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[.,[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ? = 2
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 2
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 2
[.,[[.,[[[.,.],.],.]],.]]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[.,[[[.,.],[[.,.],.]],.]]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[.,[[[.,[.,.]],[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

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

St000307The number of rowmotion orbits of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000640The rank of the largest boolean interval in a poset. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.