- FindStat

Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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

Description

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

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

Matching statistic: St000920
(load all 5 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%

Values

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

Description

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

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

Matching statistic: St000455

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000397
(load all 3 compositions to match this statistic)

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: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%

Values

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

Description

The Strahler number of a rooted tree.

Matching statistic: St001174
(load all 17 compositions to match this statistic)

Mapping

Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%

Values

[[]]
 => [.,.]
 => [1] => ? = 1 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => 0 = 1 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1 = 2 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 0 = 1 - 1
[[[],[]]]
 => [[.,.],[.,.]]
 => [1,3,2] => 0 = 1 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => 0 = 1 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0 = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1 = 2 - 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1 = 2 - 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 1 = 2 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1 = 2 - 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0 = 1 - 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1 = 2 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0 = 1 - 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0 = 1 - 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0 = 1 - 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1 = 2 - 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0 = 1 - 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0 = 1 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0 = 1 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 1 = 2 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 1 = 2 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 1 = 2 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 1 = 2 - 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 1 = 2 - 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 1 = 2 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 1 = 2 - 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 1 = 2 - 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 1 = 2 - 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 1 = 2 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => 1 = 2 - 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => 1 = 2 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1 = 2 - 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 0 = 1 - 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 1 = 2 - 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 1 = 2 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 2 - 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 1 = 2 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 0 = 1 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 1 = 2 - 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 1 = 2 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => 1 = 2 - 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => 0 = 1 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => 0 = 1 - 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 0 = 1 - 1
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0 = 1 - 1
[[],[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ? = 1 - 1
[[],[],[],[],[],[[]]]
 => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => ? = 2 - 1
[[],[],[],[],[[]],[]]
 => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => ? = 2 - 1
[[],[],[],[],[[],[]]]
 => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [5,7,6,4,3,2,1] => ? = 2 - 1
[[],[],[],[],[[[]]]]
 => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => [5,6,7,4,3,2,1] => ? = 2 - 1
[[],[],[],[[]],[],[]]
 => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => ? = 2 - 1
[[],[],[],[[]],[[]]]
 => [.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [5,6,4,7,3,2,1] => ? = 2 - 1
[[],[],[],[[],[]],[]]
 => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ? = 2 - 1
[[],[],[],[[[]]],[]]
 => [.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [5,4,6,7,3,2,1] => ? = 2 - 1
[[],[],[],[[],[],[]]]
 => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => ? = 2 - 1
[[],[],[],[[],[[]]]]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [4,6,7,5,3,2,1] => ? = 2 - 1
[[],[],[],[[[]],[]]]
 => [.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [4,5,7,6,3,2,1] => ? = 2 - 1
[[],[],[],[[[],[]]]]
 => [.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [4,6,5,7,3,2,1] => ? = 2 - 1
[[],[],[],[[[[]]]]]
 => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => [4,5,6,7,3,2,1] => ? = 2 - 1
[[],[],[[]],[],[],[]]
 => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => ? = 2 - 1
[[],[],[[]],[],[[]]]
 => [.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [5,6,4,3,7,2,1] => ? = 2 - 1
[[],[],[[]],[[]],[]]
 => [.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [5,4,6,3,7,2,1] => ? = 2 - 1
[[],[],[[]],[[],[]]]
 => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [3,5,7,6,4,2,1] => ? = 2 - 1
[[],[],[[]],[[[]]]]
 => [.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => ? = 2 - 1
[[],[],[[],[]],[],[]]
 => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ? = 2 - 1
[[],[],[[[]]],[],[]]
 => [.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [5,4,3,6,7,2,1] => ? = 2 - 1
[[],[],[[],[]],[[]]]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ? = 2 - 1
[[],[],[[[]]],[[]]]
 => [.,[.,[[[.,[[.,.],.]],.],.]]]
 => [4,5,3,6,7,2,1] => ? = 2 - 1
[[],[],[[],[],[]],[]]
 => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [5,4,3,7,6,2,1] => ? = 2 - 1
[[],[],[[],[[]]],[]]
 => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [4,5,3,7,6,2,1] => ? = 2 - 1
[[],[],[[[]],[]],[]]
 => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [4,3,5,7,6,2,1] => ? = 2 - 1
[[],[],[[[],[]]],[]]
 => [.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [4,3,6,5,7,2,1] => ? = 2 - 1
[[],[],[[[[]]]],[]]
 => [.,[.,[[[[.,[.,.]],.],.],.]]]
 => [4,3,5,6,7,2,1] => ? = 2 - 1
[[],[],[[],[],[],[]]]
 => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [3,7,6,5,4,2,1] => ? = 2 - 1
[[],[],[[],[],[[]]]]
 => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [3,6,7,5,4,2,1] => ? = 2 - 1
[[],[],[[],[[]],[]]]
 => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [3,6,5,7,4,2,1] => ? = 2 - 1
[[],[],[[],[[],[]]]]
 => [.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [4,6,5,3,7,2,1] => ? = 2 - 1
[[],[],[[],[[[]]]]]
 => [.,[.,[[.,.],[[[.,.],.],.]]]]
 => [3,5,6,7,4,2,1] => ? = 2 - 1
[[],[],[[[]],[],[]]]
 => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [3,4,7,6,5,2,1] => ? = 2 - 1
[[],[],[[[]],[[]]]]
 => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => [3,4,6,7,5,2,1] => ? = 2 - 1
[[],[],[[[],[]],[]]]
 => [.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [3,5,4,7,6,2,1] => ? = 2 - 1
[[],[],[[[[]]],[]]]
 => [.,[.,[[[[.,.],.],.],[.,.]]]]
 => [3,4,5,7,6,2,1] => ? = 2 - 1
[[],[],[[[],[],[]]]]
 => [.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [3,6,5,4,7,2,1] => ? = 2 - 1
[[],[],[[[],[[]]]]]
 => [.,[.,[[[.,.],[[.,.],.]],.]]]
 => [3,5,6,4,7,2,1] => ? = 2 - 1
[[],[],[[[[]],[]]]]
 => [.,[.,[[[[.,.],.],[.,.]],.]]]
 => [3,4,6,5,7,2,1] => ? = 2 - 1
[[],[],[[[[],[]]]]]
 => [.,[.,[[[[.,.],[.,.]],.],.]]]
 => [3,5,4,6,7,2,1] => ? = 2 - 1
[[],[],[[[[[]]]]]]
 => [.,[.,[[[[[.,.],.],.],.],.]]]
 => [3,4,5,6,7,2,1] => ? = 2 - 1
[[],[[]],[],[],[],[]]
 => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [6,5,4,3,2,7,1] => ? = 2 - 1
[[],[[]],[],[],[[]]]
 => [.,[[.,[.,[.,[[.,.],.]]]],.]]
 => [5,6,4,3,2,7,1] => ? = 2 - 1
[[],[[]],[],[[]],[]]
 => [.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [5,4,6,3,2,7,1] => ? = 2 - 1
[[],[[]],[],[[],[]]]
 => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => ? = 2 - 1
[[],[[]],[],[[[]]]]
 => [.,[[.,[.,[[[.,.],.],.]]],.]]
 => [4,5,6,3,2,7,1] => ? = 2 - 1
[[],[[]],[[]],[],[]]
 => [.,[[.,[[.,[.,[.,.]]],.]],.]]
 => [5,4,3,6,2,7,1] => ? = 2 - 1
[[],[[]],[[]],[[]]]
 => [.,[[.,[[.,[[.,.],.]],.]],.]]
 => [4,5,3,6,2,7,1] => ? = 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: St000862

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%

Values

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

Description

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

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

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

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

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

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

Matching statistic: St001621

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%

Values

[[]]
 => [.,.]
 => ([],1)
 => ([(0,1)],2)
 => 1
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[],[[]]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[],[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[],[[],[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[],[[[]]]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[[]],[],[]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[]],[[]]]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[[],[]],[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[]]],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[],[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[[]]]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[[[]],[]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[],[]]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[[]]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[],[],[]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[],[],[],[],[[]]]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2
[[],[],[],[[]],[]]
 => [[[[[.,.],.],.],[.,.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[],[],[[],[]]]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[],[[[]]]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[]],[],[]]
 => [[[[[.,.],.],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[],[[]],[[]]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[],[[],[]],[]]
 => [[[[.,.],.],[[.,.],.]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2
[[],[],[[[]]],[]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2
[[],[],[[],[],[]]]
 => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[],[[]]]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[],[[[]],[]]]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[[],[]]]]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[[[]]]]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[[]],[],[],[]]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? = 2
[[],[[]],[],[[]]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 2
[[],[[]],[[]],[]]
 => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 2
[[],[[]],[[],[]]]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[[]],[[[]]]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[[],[]],[],[]]
 => [[[[.,.],[[.,.],.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[[[]]],[],[]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[[],[]],[[]]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[[[]]],[[]]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[[],[],[]],[]]
 => [[[.,.],[[[.,.],.],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[],[[]]],[]]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 2
[[],[[[]],[]],[]]
 => [[[.,.],[[.,[.,.]],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[[],[]]],[]]
 => [[[.,.],[.,[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[[[]]]],[]]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[],[],[],[]]]
 => [[.,.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2
[[[]],[],[],[],[]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[]],[],[],[]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[]]],[],[],[]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[],[]],[],[]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[]],[]],[],[]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[],[]]],[],[]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[[]]]],[],[]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

Matching statistic: St001624

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%

Values

[[]]
 => [.,.]
 => ([],1)
 => ([(0,1)],2)
 => 1
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[],[[]]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[],[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[],[[],[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[],[[[]]]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[[]],[],[]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[]],[[]]]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[[],[]],[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[]]],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[],[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[[]]]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[[[]],[]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[],[]]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[[]]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[],[],[]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[],[],[],[],[[]]]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2
[[],[],[],[[]],[]]
 => [[[[[.,.],.],.],[.,.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[],[],[[],[]]]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[],[[[]]]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[]],[],[]]
 => [[[[[.,.],.],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[],[[]],[[]]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[],[[],[]],[]]
 => [[[[.,.],.],[[.,.],.]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2
[[],[],[[[]]],[]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2
[[],[],[[],[],[]]]
 => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[],[[]]]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[],[[[]],[]]]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[[],[]]]]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[[[]]]]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[[]],[],[],[]]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? = 2
[[],[[]],[],[[]]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 2
[[],[[]],[[]],[]]
 => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 2
[[],[[]],[[],[]]]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[[]],[[[]]]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[[],[]],[],[]]
 => [[[[.,.],[[.,.],.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[[[]]],[],[]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[[],[]],[[]]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[[[]]],[[]]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[[],[],[]],[]]
 => [[[.,.],[[[.,.],.],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[],[[]]],[]]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 2
[[],[[[]],[]],[]]
 => [[[.,.],[[.,[.,.]],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[[],[]]],[]]
 => [[[.,.],[.,[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[[[]]]],[]]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[],[],[],[]]]
 => [[.,.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2
[[[]],[],[],[],[]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[]],[],[],[]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[]]],[],[],[]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[],[]],[],[]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[]],[]],[],[]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[],[]]],[],[]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[[]]]],[],[]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The breadth of a lattice. The '''breadth''' of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .

Matching statistic: St001878

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%

Values

[[]]
 => [.,.]
 => ([],1)
 => ([(0,1)],2)
 => ? = 1
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(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
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[],[],[[]]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[],[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[],[[],[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[],[[[]]]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[[]],[],[]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[]],[[]]]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[[[[],[]],[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[]]],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[],[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[],[[]]]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[[[[[]],[]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[],[]]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[[[[[]]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[],[],[]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[],[],[],[],[[]]]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2
[[],[],[],[[]],[]]
 => [[[[[.,.],.],.],[.,.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[],[],[[],[]]]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[],[[[]]]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[]],[],[]]
 => [[[[[.,.],.],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[],[[]],[[]]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[],[[],[]],[]]
 => [[[[.,.],.],[[.,.],.]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2
[[],[],[[[]]],[]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2
[[],[],[[],[],[]]]
 => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[],[[]]]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[],[[[]],[]]]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[[],[]]]]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[],[[[[]]]]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 2
[[],[[]],[],[],[]]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? = 2
[[],[[]],[],[[]]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? = 2
[[],[[]],[[]],[]]
 => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 2
[[],[[]],[[],[]]]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[[]],[[[]]]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2
[[],[[],[]],[],[]]
 => [[[[.,.],[[.,.],.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[[[]]],[],[]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 2
[[],[[],[]],[[]]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[[[]]],[[]]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
 => ? = 2
[[],[[],[],[]],[]]
 => [[[.,.],[[[.,.],.],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[],[[]]],[]]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? = 2
[[],[[[]],[]],[]]
 => [[[.,.],[[.,[.,.]],.]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[[],[]]],[]]
 => [[[.,.],[.,[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[],[[[[]]]],[]]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 2
[[[]],[],[],[],[]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[]],[],[],[]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[]]],[],[],[]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[],[]],[],[]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[]],[]],[],[]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[],[]]],[],[]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[[[]]]],[],[]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[[[],[],[],[]],[]]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

Matching statistic: St000805

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%

Values

[[]]
 => [1,0]
 => 10 => [1,2] => 1
[[],[]]
 => [1,0,1,0]
 => 1010 => [1,2,2] => 1
[[[]]]
 => [1,1,0,0]
 => 1100 => [1,1,3] => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 101010 => [1,2,2,2] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 101100 => [1,2,1,3] => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 110010 => [1,1,3,2] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 110100 => [1,1,2,3] => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 111000 => [1,1,1,4] => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => [1,2,2,2,2] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => [1,2,2,1,3] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => [1,2,1,3,2] => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => [1,2,1,2,3] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => [1,2,1,1,4] => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => [1,1,3,2,2] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => [1,1,3,1,3] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => [1,1,2,3,2] => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => [1,1,1,4,2] => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => [1,1,2,2,3] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => [1,1,2,1,4] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => [1,1,1,3,3] => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => [1,1,1,2,4] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => [1,1,1,1,5] => 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => [1,2,2,2,2,2] => ? = 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => [1,2,2,2,1,3] => ? = 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => [1,2,2,1,3,2] => ? = 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => [1,2,2,1,2,3] => ? = 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => [1,2,2,1,1,4] => ? = 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => [1,2,1,3,2,2] => ? = 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => [1,2,1,3,1,3] => ? = 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => [1,2,1,2,3,2] => ? = 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => [1,2,1,1,4,2] => ? = 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => [1,2,1,2,2,3] => ? = 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => [1,2,1,2,1,4] => ? = 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => [1,2,1,1,3,3] => ? = 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => [1,2,1,1,2,4] => ? = 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => [1,2,1,1,1,5] => ? = 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => [1,1,3,2,2,2] => ? = 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => [1,1,3,2,1,3] => ? = 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => [1,1,3,1,3,2] => ? = 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => [1,1,3,1,2,3] => ? = 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => [1,1,3,1,1,4] => ? = 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => [1,1,2,3,2,2] => ? = 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => [1,1,1,4,2,2] => ? = 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => [1,1,2,3,1,3] => ? = 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => [1,1,1,4,1,3] => ? = 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => [1,1,2,2,3,2] => ? = 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => [1,1,2,1,4,2] => ? = 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => [1,1,1,3,3,2] => ? = 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => [1,1,1,2,4,2] => ? = 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => [1,1,1,1,5,2] => ? = 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => [1,1,2,2,2,3] => ? = 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => [1,1,2,2,1,4] => ? = 2
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => [1,1,2,1,3,3] => ? = 2
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => [1,1,2,1,2,4] => ? = 2
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => [1,1,2,1,1,5] => ? = 2
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => [1,1,1,3,2,3] => ? = 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => [1,1,1,3,1,4] => ? = 2
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => [1,1,1,2,3,3] => ? = 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => [1,1,1,1,4,3] => ? = 1
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [1,1,1,2,2,4] => ? = 1
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => [1,1,1,2,1,5] => ? = 2
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => [1,1,1,1,3,4] => ? = 1
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => [1,1,1,1,2,5] => ? = 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 1
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => [1,2,2,2,2,2,2] => ? = 1
[[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => [1,2,2,2,2,1,3] => ? = 2
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => [1,2,2,2,1,3,2] => ? = 2
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => [1,2,2,2,1,2,3] => ? = 2
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => [1,2,2,2,1,1,4] => ? = 2
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => [1,2,2,1,3,2,2] => ? = 2
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => [1,2,2,1,3,1,3] => ? = 2
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => [1,2,2,1,2,3,2] => ? = 2

Description

The number of peaks of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.

The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001729The number of visible descents of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000454The largest eigenvalue of a graph if it is integral.