Loading [MathJax]/jax/output/HTML-CSS/jax.js

Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000121
(load all 10 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000121: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [.,.]
 => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [[.,[.,.]],.]
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
 => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
 => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
 => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
 => 1 = 2 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,.]]]]}}} in a binary tree. [[oeis:A036765]] counts binary trees avoiding this pattern.
Matching statistic: St001063
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [.,.]
 => [1,0]
 => 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
Description
Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.
Matching statistic: St001064
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [.,.]
 => [1,0]
 => 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
Description
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.
Matching statistic: St001719
(load all 23 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[.,.]
 => [1] => [1] => ([(0,1)],2)
 => 1
[.,[.,.]]
 => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[.,.],.]
 => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 1
[[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
 => ? = 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 2
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
 => ? = 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 2
[[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
[[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1
[[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
[[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
 => ? = 2
[[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 2
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 3
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [4,3,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => [5,3,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,10),(4,9),(5,8),(6,11),(7,9),(7,10),(9,12),(10,12),(11,8),(12,11)],13)
 => ? = 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ? = 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => [3,4,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => ? = 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [5,6,3,4,2,1] => [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [6,4,3,5,2,1] => [3,5,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [6,3,4,5,2,1] => [5,4,3,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
 => ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [3,6,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [3,6,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [5,3,4,6,2,1] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
 => ? = 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [6,5,4,2,3,1] => [4,2,5,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
 => ? = 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [5,6,4,2,3,1] => [4,2,6,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [6,4,5,2,3,1] => [5,2,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [5,4,6,2,3,1] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
 => ? = 1
[.,[[.,.],[[[.,.],.],.]]]
 => [4,5,6,2,3,1] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [6,5,3,2,4,1] => [3,5,2,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
 => ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [5,6,3,2,4,1] => [3,6,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [6,5,2,3,4,1] => [5,2,3,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => ? = 1
[[.,.],[.,[.,[[.,.],.]]]]
 => [5,6,4,3,1,2] => [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[[.,.],[.,[[[.,.],.],.]]]
 => [4,5,6,3,1,2] => [6,3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
[[.,.],[[.,.],[[.,.],.]]]
 => [5,6,3,4,1,2] => [3,4,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[[.,.],[[.,[.,[.,.]]],.]]
 => [5,4,3,6,1,2] => [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[.,.],[[.,[[.,.],.]],.]]
 => [4,5,3,6,1,2] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[[.,[.,.]],[.,[.,[.,.]]]]
 => [6,5,4,2,1,3] => [4,2,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[.,[.,.]],[.,[[.,.],.]]]
 => [5,6,4,2,1,3] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[.,[.,.]],[[.,.],[.,.]]]
 => [6,4,5,2,1,3] => [5,2,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[.,[.,.]],[[.,[.,.]],.]]
 => [5,4,6,2,1,3] => [6,2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
[[[.,.],.],[.,[.,[.,.]]]]
 => [6,5,4,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[[.,.],.],[.,[[.,.],.]]]
 => [5,6,4,1,2,3] => [4,1,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[[.,.],.],[[.,.],[.,.]]]
 => [6,4,5,1,2,3] => [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[.,[.,[.,.]]],[.,[.,.]]]
 => [6,5,3,2,1,4] => [3,5,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[.,[.,[.,.]]],[[.,.],.]]
 => [5,6,3,2,1,4] => [3,6,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001720
(load all 23 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[.,.]
 => [1] => [1] => ([(0,1)],2)
 => 2 = 1 + 1
[.,[.,.]]
 => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[.,.],.]
 => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 1 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 1 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
 => ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 2 + 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
 => ? = 1 + 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 2 + 1
[[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 1 + 1
[[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 1 + 1
[[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
 => ? = 2 + 1
[[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 1
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 2 + 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 3 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [4,3,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => [5,3,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,10),(4,9),(5,8),(6,11),(7,9),(7,10),(9,12),(10,12),(11,8),(12,11)],13)
 => ? = 1 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ? = 1 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => [3,4,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => ? = 1 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [5,6,3,4,2,1] => [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [6,4,3,5,2,1] => [3,5,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [6,3,4,5,2,1] => [5,4,3,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
 => ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [3,6,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 2 = 1 + 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [3,6,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [5,3,4,6,2,1] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
 => ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ? = 2 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [6,5,4,2,3,1] => [4,2,5,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
 => ? = 1 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [5,6,4,2,3,1] => [4,2,6,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 2 = 1 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [6,4,5,2,3,1] => [5,2,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [5,4,6,2,3,1] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
 => ? = 1 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => [4,5,6,2,3,1] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [6,5,3,2,4,1] => [3,5,2,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
 => ? = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [5,6,3,2,4,1] => [3,6,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [6,5,2,3,4,1] => [5,2,3,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => ? = 1 + 1
[[.,.],[.,[.,[[.,.],.]]]]
 => [5,6,4,3,1,2] => [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 2 = 1 + 1
[[.,.],[.,[[[.,.],.],.]]]
 => [4,5,6,3,1,2] => [6,3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 2 = 1 + 1
[[.,.],[[.,.],[[.,.],.]]]
 => [5,6,3,4,1,2] => [3,4,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 2 = 1 + 1
[[.,.],[[.,[.,[.,.]]],.]]
 => [5,4,3,6,1,2] => [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[.,.],[[.,[[.,.],.]],.]]
 => [4,5,3,6,1,2] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 2 = 1 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
 => [6,5,4,2,1,3] => [4,2,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[.,[.,.]],[.,[[.,.],.]]]
 => [5,6,4,2,1,3] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],[.,.]]]
 => [6,4,5,2,1,3] => [5,2,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[.,[.,.]],[[.,[.,.]],.]]
 => [5,4,6,2,1,3] => [6,2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 2 = 1 + 1
[[[.,.],.],[.,[.,[.,.]]]]
 => [6,5,4,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[[.,.],.],[.,[[.,.],.]]]
 => [5,6,4,1,2,3] => [4,1,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[[.,.],.],[[.,.],[.,.]]]
 => [6,4,5,1,2,3] => [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,[.,.]]]
 => [6,5,3,2,1,4] => [3,5,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[[.,[.,[.,.]]],[[.,.],.]]
 => [5,6,3,2,1,4] => [3,6,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
Description
The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001487
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[.,.]
 => [1,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,.]]
 => [1,1,0,0]
 => []
 => [[],[]]
 => ? = 1
[[.,.],.]
 => [1,0,1,0]
 => [1]
 => [[1],[]]
 => 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1]
 => [[1],[]]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[3],[]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[3,1],[]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 2
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ? = 3
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[2],[]]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1
[.,[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1
[.,[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[4],[]]
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[4,1],[]]
 => 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1
[.,[[[.,.],[[.,.],.]],.]]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 15 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[.,.]
 => [1,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,.]]
 => [1,1,0,0]
 => []
 => [[],[]]
 => ? = 1
[[.,.],.]
 => [1,0,1,0]
 => [1]
 => [[1],[]]
 => 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1]
 => [[1],[]]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[3],[]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[3,1],[]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 2
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ? = 3
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[2],[]]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1
[.,[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1
[.,[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[4],[]]
 => 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[4,1],[]]
 => 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1
[.,[[[.,.],[[.,.],.]],.]]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 1
Description
The number of connected components of a skew partition.
Matching statistic: St001435
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[.,.]
 => [1,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,.]]
 => [1,1,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[[.,.],.]
 => [1,0,1,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[3],[]]
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[3,1],[]]
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2 - 1
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1 - 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1 - 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1 - 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 1 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2 - 1
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 1 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 2 - 1
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ? = 3 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1 - 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1 - 1
[.,[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[4],[]]
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[4,1],[]]
 => 0 = 1 - 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2 - 1
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1 - 1
[.,[[[.,.],[[.,.],.]],.]]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 1 - 1
Description
The number of missing boxes in the first row.
Matching statistic: St001438
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[.,.]
 => [1,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,.]]
 => [1,1,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[[.,.],.]
 => [1,0,1,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[3],[]]
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[3,1],[]]
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1 - 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2 - 1
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1 - 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1 - 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1 - 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 1 - 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 1 - 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2 - 1
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 1 - 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 2 - 1
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ? = 3 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => [[],[]]
 => ? = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [[1],[]]
 => 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[2],[]]
 => 0 = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0 = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 0 = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0 = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 2 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ? = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ? = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ? = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 1 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 1 - 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 1 - 1
[.,[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [[3,3,2,1],[]]
 => ? = 2 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[4],[]]
 => 0 = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[4,1],[]]
 => 0 = 1 - 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 2 - 1
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 1 - 1
[.,[[[.,.],[[.,.],.]],.]]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 1 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 1 - 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St001520
(load all 2 compositions to match this statistic)
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 60%
Values
[.,.]
 => [1,0]
 => [[1]]
 => [1] => ? = 1 - 1
[.,[.,.]]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0 = 1 - 1
[[.,.],.]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 0 = 1 - 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [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
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [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
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [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 - 1
[[.,.],[[.,.],.]]
 => [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 - 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 - 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [3,2,1,4] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [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 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,4,3,2] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[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]]
 => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [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
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0 = 1 - 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
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [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
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [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
[.,[[.,[[.,.],.]],.]]
 => [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],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [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 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 0 = 1 - 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 - 1
[[.,.],[[[.,.],.],.]]
 => [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],[0,0,0,1,0],[0,0,1,0,0]]
 => [2,1,5,4,3] => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 0 = 1 - 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 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,2,1,4,5] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,2,1,5,4] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [4,3,2,1,5] => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
 => [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 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [1,2,3,4,5,6] => ? = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [1,2,3,4,6,5] => ? = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [1,2,3,5,4,6] => ? = 1 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [1,2,3,4,6,5] => ? = 1 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [1,2,3,6,5,4] => ? = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [1,2,4,3,5,6] => ? = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [1,2,4,3,6,5] => ? = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [1,2,3,5,4,6] => ? = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => [1,2,5,4,3,6] => ? = 1 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [1,2,3,4,6,5] => ? = 1 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [1,2,3,6,5,4] => ? = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [1,2,4,3,6,5] => ? = 1 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => [1,2,3,6,5,4] => ? = 1 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => ? = 2 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[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]]
 => [1,3,2,4,5,6] => ? = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [1,3,2,4,6,5] => ? = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [1,3,2,5,4,6] => ? = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [1,3,2,4,6,5] => ? = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [1,3,2,6,5,4] => ? = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [1,2,4,3,5,6] => ? = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [1,2,4,3,6,5] => ? = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [1,4,3,2,5,6] => ? = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [1,4,3,2,6,5] => ? = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [1,2,3,5,4,6] => ? = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => [1,2,5,4,3,6] => ? = 1 - 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [1,3,2,5,4,6] => ? = 1 - 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => [1,2,5,4,3,6] => ? = 1 - 1
[.,[[[[.,.],.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => [1,5,4,3,2,6] => ? = 2 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [1,2,3,4,6,5] => ? = 1 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [1,2,3,6,5,4] => ? = 1 - 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [1,2,4,3,6,5] => ? = 1 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => [1,2,3,6,5,4] => ? = 1 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => ? = 2 - 1
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [1,3,2,4,6,5] => ? = 1 - 1
[.,[[[.,.],[[.,.],.]],.]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [1,3,2,6,5,4] => ? = 1 - 1
[.,[[[.,[.,.]],[.,.]],.]]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => [1,2,4,3,6,5] => ? = 1 - 1
[.,[[[[.,.],.],[.,.]],.]]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => [1,4,3,2,6,5] => ? = 1 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => [1,2,3,6,5,4] => ? = 1 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => ? = 2 - 1
[.,[[[[.,.],[.,.]],.],.]]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => [1,3,2,6,5,4] => ? = 1 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => ? = 2 - 1
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,6,5,4,3,2] => ? = 3 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[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],[0,0,0,0,0,1]]
 => [2,1,3,4,5,6] => ? = 1 - 1
[[.,.],[.,[.,[[.,.],.]]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [2,1,3,4,6,5] => ? = 1 - 1
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [2,1,3,5,4,6] => ? = 1 - 1
[[.,.],[.,[[.,[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => [2,1,3,4,6,5] => ? = 1 - 1
[[.,.],[.,[[[.,.],.],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [2,1,3,6,5,4] => ? = 1 - 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [2,1,4,3,5,6] => ? = 1 - 1
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [2,1,4,3,6,5] => ? = 1 - 1
Description
The number of strict 3-descents. A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001549The number of restricted non-inversions between exceedances. St001570The minimal number of edges to add to make a graph Hamiltonian. St001811The Castelnuovo-Mumford regularity of a permutation. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000445The number of rises of length 1 of a Dyck path. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice.