- FindStat

Your data matches 50 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000679

Mapping

St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => 1 = 2 - 1
[[],[]]
 => 1 = 2 - 1
[[[]]]
 => 1 = 2 - 1
[[],[],[]]
 => 1 = 2 - 1
[[],[[]]]
 => 1 = 2 - 1
[[[]],[]]
 => 1 = 2 - 1
[[[],[]]]
 => 2 = 3 - 1
[[[[]]]]
 => 1 = 2 - 1
[[],[],[],[]]
 => 1 = 2 - 1
[[],[],[[]]]
 => 1 = 2 - 1
[[],[[]],[]]
 => 1 = 2 - 1
[[],[[],[]]]
 => 2 = 3 - 1
[[],[[[]]]]
 => 1 = 2 - 1
[[[]],[],[]]
 => 1 = 2 - 1
[[[]],[[]]]
 => 1 = 2 - 1
[[[],[]],[]]
 => 2 = 3 - 1
[[[[]]],[]]
 => 1 = 2 - 1
[[[],[],[]]]
 => 2 = 3 - 1
[[[],[[]]]]
 => 2 = 3 - 1
[[[[]],[]]]
 => 2 = 3 - 1
[[[[],[]]]]
 => 2 = 3 - 1
[[[[[]]]]]
 => 1 = 2 - 1
[[],[],[],[],[]]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => 2 = 3 - 1
[[],[],[[[]]]]
 => 1 = 2 - 1
[[],[[]],[],[]]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => 2 = 3 - 1
[[],[[[]]],[]]
 => 1 = 2 - 1
[[],[[],[],[]]]
 => 2 = 3 - 1
[[],[[],[[]]]]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => 1 = 2 - 1
[[[]],[],[],[]]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => 1 = 2 - 1
[[[],[]],[],[]]
 => 2 = 3 - 1
[[[[]]],[],[]]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => 1 = 2 - 1
[[[],[],[]],[]]
 => 2 = 3 - 1
[[[],[[]]],[]]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => 1 = 2 - 1

Description

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

$[^r]^r$ for some

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

$[]$ .

Matching statistic: St000396
(load all 3 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [.,.]
 => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 1 = 2 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1 = 2 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1 = 2 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 3 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 2 - 1

Description

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

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

Matching statistic: St000920
(load all 7 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to 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 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 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: St000397

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [.,.]
 => [[],[]]
 => 2
[[],[]]
 => [[.,.],.]
 => [[[],[]],[]]
 => 2
[[[]]]
 => [.,[.,.]]
 => [[],[[],[]]]
 => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2
[[],[[]]]
 => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 3
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 3
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 3
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 3
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 3
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [[[[[[],[]],[]],[]],[]],[]]
 => 2
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[[],[]],[]],[]],[[],[]]]
 => 3
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [[[[[],[]],[]],[[],[]]],[]]
 => 3
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => 3
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => 3
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [[[[[],[]],[[],[]]],[]],[]]
 => 3
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[[],[]],[[],[]]],[[],[]]]
 => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [[[[],[]],[[[],[]],[]]],[]]
 => 3
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [[[[],[]],[[],[[],[]]]],[]]
 => 3
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => 3
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => 3
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => 3
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => 3
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => 3
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[[],[]]],[]],[]],[]]
 => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 3
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[[],[]]],[[],[]]],[]]
 => 3
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 3
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 3
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[[],[]],[]]],[]],[]]
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[[],[[],[]]]],[]],[]]
 => 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 3
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => 3
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[[],[]],[]],[]]],[]]
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[[],[]],[[],[]]]],[]]
 => 3
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[[],[[],[]]],[]]],[]]
 => 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[[],[[[],[]],[]]]],[]]
 => 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[[],[[],[[],[]]]]],[]]
 => 2

Description

The Strahler number of a rooted tree.

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

Mapping

Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000862
(load all 3 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [.,.]
 => [1] => [1] => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => [1,2] => [1,2] => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => 1 = 2 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => 1 = 2 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => 1 = 2 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => 2 = 3 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => 2 = 3 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => 1 = 2 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => 2 = 3 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [1,2,3,5,4] => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [1,2,4,5,3] => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [1,2,5,3,4] => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [1,2,5,3,4] => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [1,3,4,5,2] => 2 = 3 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [1,3,5,2,4] => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [1,4,5,2,3] => 2 = 3 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [1,4,5,2,3] => 2 = 3 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [1,5,2,3,4] => 2 = 3 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [1,5,2,3,4] => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [1,5,2,4,3] => 2 = 3 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [1,5,2,3,4] => 2 = 3 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [1,5,2,3,4] => 2 = 3 - 1

Description

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

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

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

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

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

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

Matching statistic: St001235

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St001261

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Castelnuovo-Mumford regularity of a graph.

Matching statistic: St001741

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [.,.]
 => [1] => [1] => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => [1,2] => [1,2] => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => 1 = 2 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => 1 = 2 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => 1 = 2 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => 1 = 2 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => 2 = 3 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => 2 = 3 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => 1 = 2 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => 2 = 3 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [1,2,3,5,4] => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [1,2,4,5,3] => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [1,2,5,3,4] => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [1,2,5,3,4] => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [1,3,4,5,2] => 2 = 3 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [1,3,5,2,4] => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [1,4,5,2,3] => 2 = 3 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [1,4,5,2,3] => 2 = 3 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [1,5,2,3,4] => 2 = 3 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [1,5,2,3,4] => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [1,5,2,4,3] => 2 = 3 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [1,5,2,3,4] => 2 = 3 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [1,5,2,3,4] => 2 = 3 - 1

Description

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

Matching statistic: St000535

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rank-width of a graph.

The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000660The number of rises of length at least 3 of a Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001393The induced matching number of a graph. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St000640The rank of the largest boolean interval in a poset. St000307The number of rowmotion orbits of a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000481The number of upper covers of a partition in dominance order. St001330The hat guessing number of a graph. St000480The number of lower covers of a partition in dominance order. 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

$eA$ . St001820The size of the image of the pop stack sorting operator. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001846The number of elements which do not have a complement in the lattice. St001964The interval resolution global dimension of a poset. St001626The number of maximal proper sublattices of a lattice. St000897The number of different multiplicities of parts of an integer partition. St000031The number of cycles in the cycle decomposition of a permutation. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation. St000761The number of ascents in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001960The number of descents of a permutation minus one if its first entry is not one.