- FindStat

Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000118
(load all 11 compositions to match this statistic)

Mapping

St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree. [[oeis:A001006]] counts binary trees avoiding this pattern.

Matching statistic: St001066
(load all 10 compositions to match this statistic)

Mapping

Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [.,.]
 => [1,0]
 => 1 = 0 + 1
[.,[.,.]]
 => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[.,.],.]
 => [[.,.],.]
 => [1,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],.]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1

Description

The number of simple reflexive modules in the corresponding Nakayama algebra.

Matching statistic: St000931
(load all 11 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern UUU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Matching statistic: St001167

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. The top of a module is the cokernel of the inclusion of the radical of the module into the module. For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001253

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. For the first 196 values the statistic coincides also with the number of fixed points of

$\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details. The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).

Matching statistic: St001483
(load all 2 compositions to match this statistic)

Mapping

Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [.,.]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[.,[.,.]]
 => [[.,.],.]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[[.,.],.]
 => [.,[.,.]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],.],.]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1

Description

The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.

Matching statistic: St000366
(load all 16 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double descents of a permutation. A double descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) > \pi(i+1) > \pi(i+2)$ .

Matching statistic: St000371
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) > \pi(j) > \pi(k)$ . In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

Matching statistic: St000358
(load all 2 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$31\!\!-\!\!2$ .

Matching statistic: St001727

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of invisible inversions of a permutation. A visible inversion of a permutation

$\pi$ is a pair

$i < j$ such that

$\pi(j) \leq \min(i, \pi(i))$ . Thus, an invisible inversion satisfies

$\pi(i) > \pi(j) > i$ .

The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000836The number of descents of distance 2 of a permutation. St000732The number of double deficiencies of a permutation. St000365The number of double ascents of a permutation. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000223The number of nestings in the permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St000982The length of the longest constant subword.