- FindStat

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

Matching statistic: St000018
(load all 2 compositions to match this statistic)

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
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => []
 => [1,0]
 => [1] => 0
[[]]
 => [1,0]
 => [1,1,0,0]
 => [2,1] => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 10
[[],[],[],[],[]]
 => [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] => 5
[[],[],[],[[]]]
 => [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] => 6
[[],[],[[]],[]]
 => [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] => 6
[[],[],[[],[]]]
 => [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] => 7
[[],[],[[[]]]]
 => [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] => 8
[[],[[]],[],[]]
 => [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] => 6
[[],[[]],[[]]]
 => [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] => 7
[[],[[],[]],[]]
 => [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] => 7
[[],[[[]]],[]]
 => [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] => 8
[[],[[],[],[]]]
 => [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] => 8
[[],[[],[[]]]]
 => [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] => 9
[[],[[[]],[]]]
 => [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] => 9
[[],[[[],[]]]]
 => [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] => 10
[[],[[[[]]]]]
 => [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] => 11
[[[]],[],[],[]]
 => [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] => 6
[[[]],[],[[]]]
 => [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] => 7
[[[]],[[]],[]]
 => [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] => 7
[[[]],[[],[]]]
 => [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] => 8
[[[]],[[[]]]]
 => [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] => 9
[[[],[]],[],[]]
 => [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] => 7
[[[[]]],[],[]]
 => [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] => 8
[[[],[]],[[]]]
 => [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] => 8
[[[[]]],[[]]]
 => [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] => 9
[[[],[],[]],[]]
 => [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] => 8
[[[],[[]]],[]]
 => [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] => 9
[[[[]],[]],[]]
 => [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] => 9
[[[[],[]]],[]]
 => [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] => 10

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

Matching statistic: St000400
(load all 2 compositions to match this statistic)

Mapping

St000400: Ordered trees ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%

Values

[]
 => ? = 0
[[]]
 => 1
[[],[]]
 => 2
[[[]]]
 => 3
[[],[],[]]
 => 3
[[],[[]]]
 => 4
[[[]],[]]
 => 4
[[[],[]]]
 => 5
[[[[]]]]
 => 6
[[],[],[],[]]
 => 4
[[],[],[[]]]
 => 5
[[],[[]],[]]
 => 5
[[],[[],[]]]
 => 6
[[],[[[]]]]
 => 7
[[[]],[],[]]
 => 5
[[[]],[[]]]
 => 6
[[[],[]],[]]
 => 6
[[[[]]],[]]
 => 7
[[[],[],[]]]
 => 7
[[[],[[]]]]
 => 8
[[[[]],[]]]
 => 8
[[[[],[]]]]
 => 9
[[[[[]]]]]
 => 10
[[],[],[],[],[]]
 => 5
[[],[],[],[[]]]
 => 6
[[],[],[[]],[]]
 => 6
[[],[],[[],[]]]
 => 7
[[],[],[[[]]]]
 => 8
[[],[[]],[],[]]
 => 6
[[],[[]],[[]]]
 => 7
[[],[[],[]],[]]
 => 7
[[],[[[]]],[]]
 => 8
[[],[[],[],[]]]
 => 8
[[],[[],[[]]]]
 => 9
[[],[[[]],[]]]
 => 9
[[],[[[],[]]]]
 => 10
[[],[[[[]]]]]
 => 11
[[[]],[],[],[]]
 => 6
[[[]],[],[[]]]
 => 7
[[[]],[[]],[]]
 => 7
[[[]],[[],[]]]
 => 8
[[[]],[[[]]]]
 => 9
[[[],[]],[],[]]
 => 7
[[[[]]],[],[]]
 => 8
[[[],[]],[[]]]
 => 8
[[[[]]],[[]]]
 => 9
[[[],[],[]],[]]
 => 8
[[[],[[]]],[]]
 => 9
[[[[]],[]],[]]
 => 9
[[[[],[]]],[]]
 => 10
[[[[[]]]],[]]
 => 11
[[[[[[[[[]]]]]]]]]
 => ? = 36

Description

The path length of an ordered tree. This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].

Matching statistic: St000161

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 68% ●values known / values provided: 85%●distinct values known / distinct values provided: 68%

Values

[]
 => []
 => [1,0]
 => [.,.]
 => 0
[[]]
 => [1,0]
 => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => 2
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => 6
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [.,[[[[.,.],.],[.,.]],.]]
 => 6
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => 7
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => 8
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [.,[[[[.,.],[.,.]],.],.]]
 => 6
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => 7
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [.,[[[.,.],[[.,.],.]],.]]
 => 7
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [.,[[[.,.],[.,[.,.]]],.]]
 => 8
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [.,[[.,.],[[[.,.],.],.]]]
 => 8
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => 9
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => 9
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => 10
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => 11
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => 6
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => 7
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [.,[[[.,[.,.]],[.,.]],.]]
 => 7
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => 8
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => 9
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => 7
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => 8
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => 8
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => 9
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => 8
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => 9
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => 9
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => 10
[[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ? = 21
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
 => ? = 12
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
 => ? = 20
[[[]],[[[]]],[[]]]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
 => ? = 12
[[[[[[]]]],[]],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
 => ? = 18
[[[[[[]],[]]]],[]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
 => ? = 20
[[[[[[[]]]]]],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
 => ? = 22
[[[],[[],[[[]]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
 => ? = 20
[[[[]],[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
 => ? = 16
[[[[[[]],[]],[]]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
 => ? = 22
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
 => ? = 24
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
 => ? = 24
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
 => ? = 26
[[[[[[[[]]]]]]]]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 28
[[[[[[[[[]]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => ? = 36

Description

The sum of the sizes of the right subtrees of a binary tree. This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the $312$-avoiding permutation corresponding to the binary tree. It is also the sum of all heights $j$ of the coordinates $(i,j)$ of the Dyck path corresponding to the binary tree.

Matching statistic: St000041

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 68% ●values known / values provided: 79%●distinct values known / distinct values provided: 68%

Values

[]
 => []
 => [1,0]
 => [(1,2)]
 => 0
[[]]
 => [1,0]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => 10
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
 => 6
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]
 => 6
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
 => 7
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
 => 8
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]
 => 6
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
 => 7
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)]
 => 7
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
 => 8
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)]
 => 8
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,6),(7,10),(8,9)]
 => 9
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)]
 => 9
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]
 => 10
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
 => 11
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => 6
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]
 => 7
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]
 => 7
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,8),(9,10)]
 => 8
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
 => 9
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => 7
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => 8
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,11),(9,10)]
 => 8
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,11),(9,10)]
 => 9
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
 => 8
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,4),(5,8),(6,7),(10,11)]
 => 9
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [(1,12),(2,9),(3,6),(4,5),(7,8),(10,11)]
 => 9
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
 => 10
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [(1,16),(2,3),(4,5),(6,15),(7,8),(9,10),(11,14),(12,13)]
 => ? = 12
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [(1,16),(2,3),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
 => ? = 20
[[[]],[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [(1,16),(2,5),(3,4),(6,9),(7,8),(10,13),(11,12),(14,15)]
 => ? = 10
[[[]],[[[]]],[[]]]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [(1,16),(2,5),(3,4),(6,11),(7,10),(8,9),(12,15),(13,14)]
 => ? = 12
[[[]],[[[]],[]],[]]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [(1,16),(2,5),(3,4),(6,13),(7,10),(8,9),(11,12),(14,15)]
 => ? = 12
[[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [(1,16),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13),(14,15)]
 => ? = 10
[[[[]],[]],[[]],[]]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [(1,16),(2,9),(3,6),(4,5),(7,8),(10,13),(11,12),(14,15)]
 => ? = 12
[[[[]],[[]],[]],[]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [(1,16),(2,13),(3,6),(4,5),(7,10),(8,9),(11,12),(14,15)]
 => ? = 14
[[[[[]],[]],[]],[]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [(1,16),(2,13),(3,10),(4,7),(5,6),(8,9),(11,12),(14,15)]
 => ? = 16
[[[[[[]]]],[]],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [(1,16),(2,13),(3,10),(4,9),(5,8),(6,7),(11,12),(14,15)]
 => ? = 18
[[[[[[]],[]]]],[]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [(1,16),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(14,15)]
 => ? = 20
[[[[[[[]]]]]],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [(1,16),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,15)]
 => ? = 22
[[[],[[],[[[]]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [(1,16),(2,15),(3,4),(5,14),(6,7),(8,13),(9,12),(10,11)]
 => ? = 20
[[[[]],[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [(1,16),(2,15),(3,6),(4,5),(7,10),(8,9),(11,14),(12,13)]
 => ? = 16
[[[[[[]],[]],[]]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [(1,16),(2,15),(3,14),(4,11),(5,8),(6,7),(9,10),(12,13)]
 => ? = 22
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [(1,16),(2,15),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
 => ? = 24
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [(1,16),(2,15),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
 => ? = 24
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [(1,16),(2,15),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
 => ? = 26
[[[[[[[[]]]]]]]]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => ? = 28
[[[],[]],[],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,18),(2,7),(3,4),(5,6),(8,9),(10,11),(12,13),(14,15),(16,17)]
 => ? = 10
[[[[[[[[[]]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
 => ? = 36

Description

The number of nestings of a perfect matching. This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$.

Matching statistic: St000639

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 68% ●values known / values provided: 79%●distinct values known / distinct values provided: 68%

Values

[]
 => .
 => ? => ?
 => ? = 0
[[]]
 => [.,.]
 => [1] => ([],1)
 => ? = 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([],2)
 => 2
[[[]]]
 => [[.,.],.]
 => [1,2] => ([(0,1)],2)
 => 3
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([],3)
 => 3
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(1,2)],3)
 => 4
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(1,2)],3)
 => 4
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 5
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 6
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([],4)
 => 4
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(2,3)],4)
 => 5
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(2,3)],4)
 => 5
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 6
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 7
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(2,3)],4)
 => 5
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 6
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 7
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 7
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 8
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 8
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 9
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 10
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([],5)
 => 5
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(3,4)],5)
 => 6
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(3,4)],5)
 => 6
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 7
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 8
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(3,4)],5)
 => 6
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 7
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 7
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 8
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 8
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 9
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 9
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 10
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 11
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 6
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 7
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 7
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 8
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 9
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 7
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 8
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => 8
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 9
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 8
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 9
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 9
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 10
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 11
[[[],[],[],[]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 9
[[],[[[[],[[]]]]]]
 => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 20
[[[]],[[]],[[]],[]]
 => [[.,.],[[.,.],[[.,.],[.,.]]]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 10
[[[]],[[[]]],[[]]]
 => [[.,.],[[[.,.],.],[[.,.],.]]]
 => [6,7,3,4,5,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 12
[[[]],[[[]],[]],[]]
 => [[.,.],[[[.,.],[.,.]],[.,.]]]
 => [7,5,3,4,6,1,2] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 12
[[[[]],[]],[[]],[]]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => [7,5,6,3,1,2,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 12
[[[[]],[[]],[]],[]]
 => [[[.,.],[[.,.],[.,.]]],[.,.]]
 => [7,5,3,4,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 14
[[[[[]],[]],[]],[]]
 => [[[[.,.],[.,.]],[.,.]],[.,.]]
 => [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? = 16
[[[[[[]],[]]]],[]]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 20
[[[[[[[]]]]]],[]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 22
[[[],[[],[[[]]]]]]
 => [[.,[[.,[[[.,.],.],.]],.]],.]
 => [3,4,5,2,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ? = 20
[[[[]],[[]],[[]]]]
 => [[[.,.],[[.,.],[[.,.],.]]],.]
 => [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 16
[[[[[[]],[]],[]]]]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ? = 22
[[[[[[]],[[]]]]]]
 => [[[[[.,.],[[.,.],.]],.],.],.]
 => [3,4,1,2,5,6,7] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 24
[[[[[[[]],[]]]]]]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 26
[[[[[[[[]]]]]]]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 28
[[],[],[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [8,7,6,5,4,3,2,1] => ([],8)
 => ? = 8
[[[]],[],[],[],[],[],[]]
 => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 9
[[[],[]],[],[],[],[],[]]
 => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => [8,7,6,5,4,2,1,3] => ([(5,7),(6,7)],8)
 => ? = 10
[[[[[[[[[]]]]]]]]]
 => [[[[[[[[.,.],.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 36

Description

The number of relations in a poset. This is the number of intervals $x,y$ with $x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.

Matching statistic: St000081

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 76%

Values

[]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 6
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 7
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 10
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 11
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 7
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 7
[[[]],[[],[]]]
 => ([(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)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[[],[]],[[]]]
 => ([(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)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 10
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 7
[[],[],[[],[],[[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ?
 => ?
 => ? = 12
[[],[[[[],[[]]]]]]
 => ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
 => ?
 => ?
 => ? = 20
[[[]],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 8
[[[]],[[]],[[]],[]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 10
[[[]],[[[]]],[[]]]
 => ([(0,5),(1,4),(2,6),(3,7),(4,7),(5,7),(6,3)],8)
 => ?
 => ?
 => ? = 12
[[[]],[[[]],[]],[]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 12
[[[],[]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 9
[[[[]]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 10
[[[],[],[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 10
[[[[]],[]],[[]],[]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 12
[[[[]],[[]],[]],[]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(7,6)],8)
 => ?
 => ?
 => ? = 14
[[[[[]],[]],[]],[]]
 => ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
 => ?
 => ?
 => ? = 16
[[[[[[]]]],[]],[]]
 => ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
 => ?
 => ?
 => ? = 18
[[[[[[]],[]]]],[]]
 => ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
 => ?
 => ?
 => ? = 20
[[[[[[[]]]]]],[]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ?
 => ?
 => ? = 22
[[[],[[],[[[]]]]]]
 => ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
 => ?
 => ?
 => ? = 20
[[[[]],[[]],[[]]]]
 => ([(0,6),(1,5),(2,4),(4,7),(5,7),(6,7),(7,3)],8)
 => ?
 => ?
 => ? = 16
[[[[[[]],[]],[]]]]
 => ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
 => ?
 => ?
 => ? = 22
[[[[[[[]]]],[]]]]
 => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
 => ?
 => ?
 => ? = 24
[[[[[[]],[[]]]]]]
 => ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
 => ?
 => ?
 => ? = 24
[[[[[[[]],[]]]]]]
 => ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
 => ?
 => ?
 => ? = 26
[[],[],[],[],[],[],[],[]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 8
[[[]],[],[],[],[],[],[]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ?
 => ?
 => ? = 9
[[[],[]],[],[],[],[],[]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ?
 => ? = 10

Description

The number of edges of a graph.

Matching statistic: St000246

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 88%

Values

[]
 => []
 => [1,0]
 => [1] => 0
[[]]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 2
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 10
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => 6
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => 6
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => 7
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => 8
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => 6
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => 7
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => 7
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => 8
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => 8
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => 9
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => 9
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => 10
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => 11
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,2,6] => 6
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,5,3,1,2,6] => 7
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,4,1,2,6] => 7
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => 8
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6] => 9
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => 7
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,2,3,6] => 8
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => 8
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,5,1,2,3,6] => 9
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => 8
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => 9
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,2,4,6] => 9
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [5,2,1,3,4,6] => 10
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,1,2,7] => ? = 7
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2,1,3,7] => ? = 8
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => ? = 9
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2,1,4,7] => ? = 9
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3,1,2,4,7] => ? = 10
[[[],[],[],[]],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [6,4,3,2,1,5,7] => ? = 10
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [5,6,4,3,7,2,1,8] => ? = 12
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,1,8] => ? = 20
[[[]],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,1,2,8] => ? = 8
[[[]],[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,1,2,8] => ? = 10
[[[]],[[[]]],[[]]]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [6,7,3,4,5,1,2,8] => ? = 12
[[[]],[[[]],[]],[]]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,6,1,2,8] => ? = 12
[[[],[]],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,2,1,3,8] => ? = 9
[[[[]]],[],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,1,2,3,8] => ? = 10
[[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [7,6,5,3,2,1,4,8] => ? = 10
[[[[]],[]],[[]],[]]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,1,2,4,8] => ? = 12
[[[[]],[[]],[]],[]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,1,2,6,8] => ? = 14
[[[[[]],[]],[]],[]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [7,5,3,1,2,4,6,8] => ? = 16
[[[[[[]]]],[]],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [7,5,1,2,3,4,6,8] => ? = 18
[[[[[[]],[]]]],[]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [7,3,1,2,4,5,6,8] => ? = 20
[[[],[[],[[[]]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,1,7,8] => ? = 20
[[[[[[]],[]],[]]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [5,3,1,2,4,6,7,8] => ? = 22
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8] => ? = 26
[[[]],[],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,1,2,9] => ? = 9
[[[],[]],[],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,2,1,3,9] => ? = 10

Description

The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.

Matching statistic: St000012
(load all 2 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 73%●distinct values known / distinct values provided: 68%

Values

[]
 => []
 => [1,0]
 => 0
[[]]
 => [1,0]
 => [1,1,0,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 10
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 6
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 6
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 7
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 8
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 6
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 7
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 7
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 8
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 9
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 9
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 10
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 11
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 6
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 7
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 7
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 8
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 9
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 7
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 8
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 8
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 9
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 8
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 9
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 9
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 10
[[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 12
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 20
[[[]],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8
[[[]],[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 10
[[[]],[[[]]],[[]]]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 12
[[[]],[[[]],[]],[]]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 12
[[[],[]],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 9
[[[[]]],[],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 10
[[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 10
[[[[]],[]],[[]],[]]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => ? = 12
[[[[]],[[]],[]],[]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => ? = 14
[[[[[]],[]],[]],[]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 16
[[[[[[]]]],[]],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => ? = 18
[[[[[[]],[]]]],[]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => ? = 20
[[[[[[[]]]]]],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 22
[[[],[[],[[[]]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 20
[[[[]],[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 16
[[[[[[]],[]],[]]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => ? = 22
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 24
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 24
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 26
[[[[[[[[]]]]]]]]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 28
[[],[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8
[[[]],[],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 9
[[[],[]],[],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 10
[[[[[[[[[]]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 36

Description

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$. 2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

Matching statistic: St001228
(load all 4 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 72%●distinct values known / distinct values provided: 64%

Values

[]
 => []
 => ? = 0
[[]]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => 2
[[[]]]
 => [1,1,0,0]
 => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 10
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 7
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 8
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 7
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 7
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 8
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 8
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 9
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 9
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 10
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 11
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 6
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 7
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 7
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 8
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 9
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 7
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 8
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 8
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 9
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 8
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 9
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 9
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 10
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 11
[[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 12
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 20
[[[]],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[[[]],[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 10
[[[]],[[[]]],[[]]]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 12
[[[]],[[[]],[]],[]]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 12
[[[],[]],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 9
[[[[]]],[],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
[[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 10
[[[[]],[]],[[]],[]]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => ? = 12
[[[[]],[[]],[]],[]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => ? = 14
[[[[[]],[]],[]],[]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 16
[[[[[[]]]],[]],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 18
[[[[[[]],[]]]],[]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 20
[[[[[[[]]]]]],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 22
[[[],[[],[[[]]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 20
[[[[]],[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 16
[[[[[[]],[]],[]]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 22
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 24
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 24
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 26
[[[[[[[[]]]]]]]]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 28
[[],[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[[[]],[],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[[[],[]],[],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[[[[[[[[[]]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 36

Description

The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.

Matching statistic: St000067

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 68%

Values

[]
 => []
 => [1,0]
 => [[1]]
 => 0
[[]]
 => [1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 2
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => 4
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => 4
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => 5
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 6
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 5
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 5
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 6
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 7
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 5
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 6
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 6
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => 7
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 7
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 8
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => 8
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 9
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 10
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => 6
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 6
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => 7
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => 8
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 6
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => 7
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 7
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => 8
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => 8
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => 9
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => 9
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => 10
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => 11
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 6
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => 7
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 7
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => 8
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => 9
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 7
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 8
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => 8
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => 9
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 8
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => 9
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 9
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => 10
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 6
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 7
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 8
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 9
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 9
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 10
[[[],[],[],[]],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 10
[[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
 => ? = 21
[[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 7
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,-1,0,1],[0,0,0,1,-1,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 12
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,0,0,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,1,-1,0,1,0],[0,0,1,-1,0,1,0,0],[0,1,-1,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 20
[[[]],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 8
[[[]],[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,-1,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,-1,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 10
[[[]],[[[]]],[[]]]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,-1,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
 => ? = 12
[[[]],[[[]],[]],[]]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,-1,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 12
[[[],[]],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 9
[[[[]]],[],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 10
[[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[1,-1,1,-1,1,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 10
[[[[]],[]],[[]],[]]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[1,0,-1,1,0,0,0,0],[0,0,1,0,-1,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 12
[[[[]],[[]],[]],[]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,-1,0,1,0,0],[1,0,-1,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 14
[[[[[]],[]],[]],[]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,-1,1,0,0],[0,1,0,-1,1,0,0,0],[1,0,-1,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 16
[[[[[[]]]],[]],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,-1,1,0],[1,0,0,0,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 18
[[[[[[]],[]]]],[]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,1,0,-1,1,0,0,0],[1,0,-1,1,0,0,0,0],[0,0,1,0,0,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 20
[[[[[[[]]]]]],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,-1,1],[0,0,0,0,0,0,1,0]]
 => ? = 22
[[[],[[],[[[]]]]]]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [[0,0,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[1,-1,0,1,-1,0,0,1],[0,0,1,-1,0,0,1,0],[0,1,-1,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 20
[[[[]],[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,-1,0,1,0,0],[1,0,-1,0,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,1,0,-1,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 16
[[[[[[]],[]],[]]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,1,0,-1,1,0,-1,1],[1,0,-1,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 22
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,-1,1],[0,1,0,0,0,-1,1,0],[1,0,0,0,-1,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 24
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,1,0,-1,0,1,0],[0,1,0,-1,0,1,0,0],[1,0,-1,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 24
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,1,0,-1,1,0,0,0],[1,0,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 26
[[],[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
 => ? = 8
[[[]],[],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
 => ? = 9
[[[],[]],[],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
 => ? = 10
[[[[[[[[[]]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0]]
 => ? = 36

Description

The inversion number of the alternating sign matrix. If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as $$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$ When restricted to permutation matrices, this gives the usual inversion number of the permutation.

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

St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000448The number of pairs of vertices of a graph with distance 2. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001646The number of edges that can be added without increasing the maximal degree of a graph.