Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001928
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1,2] => [1] => 0
[[.,.],.]
 => [1,2] => [2,1] => [1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => [1,2] => 0
[[.,.],[.,.]]
 => [3,1,2] => [2,1,3] => [2,1] => 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [1,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,3,2,4] => [1,3,2] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,3] => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,3] => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,2,4] => [3,1,2] => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1] => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,1,3] => [2,1,3] => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [3,1,2] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0
Description
The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000337
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1] => [1] => 0
[[.,.],.]
 => [1,2] => [1] => [1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,1] => [1,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [2,1] => [1,2] => 0
[[.,.],[.,.]]
 => [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1] => [1,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,1] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1,2] => [1,3,2] => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,1,3] => [2,3,1] => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3] => [3,2,1] => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3] => [3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
 => [8,6,7,5,4,3,2,1] => ? => ? => ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
 => [8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
 => [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 2
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
 => [7,6,8,4,5,3,2,1] => ? => ? => ? = 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
 => [8,7,5,4,6,3,2,1] => ? => ? => ? = 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
 => [8,6,5,4,7,3,2,1] => ? => ? => ? = 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
 => [8,5,6,4,7,3,2,1] => ? => ? => ? = 2
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
 => [7,5,6,4,8,3,2,1] => ? => ? => ? = 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
 => [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 2
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
 => [7,6,5,8,3,4,2,1] => ? => ? => ? = 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
 => [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 2
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
 => [7,6,8,3,4,5,2,1] => ? => ? => ? = 1
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
 => [8,7,4,5,3,6,2,1] => ? => ? => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
 => [8,5,4,3,6,7,2,1] => ? => ? => ? = 1
[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
 => [6,5,4,7,3,8,2,1] => ? => ? => ? = 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
 => [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
 => [7,6,5,3,4,8,2,1] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
 => [7,6,3,4,5,8,2,1] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 1
[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]
 => [6,5,4,3,7,8,2,1] => ? => ? => ? = 1
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
 => [5,4,3,6,7,8,2,1] => ? => ? => ? = 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
 => [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 2
[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
 => [7,6,8,5,4,2,3,1] => ? => ? => ? = 1
[.,[[.,.],[.,[[.,[.,[.,.]]],.]]]]
 => [7,6,5,8,4,2,3,1] => ? => ? => ? = 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
 => [5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => ? = 2
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
 => [8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => [1,2,4,3,6,5,7] => ? = 2
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
 => [8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => [2,1,4,3,6,5,7] => ? = 3
[.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
 => [7,6,8,4,5,2,3,1] => ? => ? => ? = 2
[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
 => [8,7,5,4,6,2,3,1] => ? => ? => ? = 2
[.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
 => [8,6,5,4,7,2,3,1] => ? => ? => ? = 2
[.,[[.,.],[[.,[[.,.],.]],[.,.]]]]
 => [8,5,6,4,7,2,3,1] => ? => ? => ? = 3
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
 => [7,6,5,4,8,2,3,1] => ? => ? => ? = 1
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
 => [7,5,6,4,8,2,3,1] => ? => ? => ? = 2
[.,[[.,.],[[.,[[.,[.,.]],.]],.]]]
 => [6,5,7,4,8,2,3,1] => ? => ? => ? = 2
[.,[[.,.],[[[[[.,.],.],.],.],.]]]
 => [4,5,6,7,8,2,3,1] => [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => ? = 3
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
 => [8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,3,5,6,4,7] => ? = 1
[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
 => [8,6,7,5,3,2,4,1] => ? => ? => ? = 2
[.,[[.,[.,.]],[.,[[.,[.,.]],.]]]]
 => [7,6,8,5,3,2,4,1] => ? => ? => ? = 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
 => [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => ? = 2
[.,[[.,[.,.]],[[.,[.,[.,.]]],.]]]
 => [7,6,5,8,3,2,4,1] => ? => ? => ? = 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
 => [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 2
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
 => [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 2
[.,[[[.,.],.],[[.,[.,[.,.]]],.]]]
 => [7,6,5,8,2,3,4,1] => ? => ? => ? = 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
 => [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 2
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
 => [7,5,6,4,3,2,8,1] => ? => ? => ? = 1
[.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
 => [6,4,5,7,3,2,8,1] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 1
[.,[[.,[[.,.],[.,[.,[.,.]]]]],.]]
 => [7,6,5,3,4,2,8,1] => ? => ? => ? = 1
[.,[[.,[[.,.],[[.,.],[.,.]]]],.]]
 => [7,5,6,3,4,2,8,1] => ? => ? => ? = 2
[.,[[.,[[.,.],[[.,[.,.]],.]]],.]]
 => [6,5,7,3,4,2,8,1] => ? => ? => ? = 2
[.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
 => [7,6,4,3,5,2,8,1] => ? => ? => ? = 1
[.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
 => [7,5,4,3,6,2,8,1] => ? => ? => ? = 1
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000628
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[.,[.,.]]
 => [1,1,0,0]
 => 1100 => 1010 => 1 = 0 + 1
[[.,.],.]
 => [1,0,1,0]
 => 1010 => 0110 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 111000 => 101010 => 1 = 0 + 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 110100 => 011010 => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 101100 => 110010 => 2 = 1 + 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 110010 => 010110 => 1 = 0 + 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 101010 => 100110 => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 10101010 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 01101010 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 11001010 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 01011010 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 10011010 => 2 = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11010010 => 2 = 1 + 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 10110010 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 00111010 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 01110010 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 01010110 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 10010110 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 10100110 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 00110110 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 01100110 => 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => ? => ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => ? => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? => ? = 1 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? => ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? => ? = 1 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => ? => ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? => ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? => ? = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? => ? = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? => ? = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? => ? = 1 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? => ? = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? => ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? => ? = 1 + 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? => ? = 2 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? => ? = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? => ? = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? => ? = 1 + 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? => ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? => ? = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? => ? = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? => ? = 1 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? => ? = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? => ? = 2 + 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? => ? = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? => ? = 1 + 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? => ? = 1 + 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? => ? = 1 + 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? => ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? => ? = 1 + 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? => ? = 2 + 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? => ? = 1 + 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? => ? = 1 + 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? => ? = 1 + 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? => ? = 2 + 1
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? => ? = 1 + 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? => ? = 1 + 1
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? => ? = 2 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 111111000000 => ? => ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => ? => ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 111101100000 => ? => ? = 1 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => ? => ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => ? => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 111011100000 => ? => ? = 1 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 111011010000 => ? => ? = 1 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 111100110000 => ? => ? = 1 + 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.