Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000542
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [1,2] => 1
[1,1,0,0]
 => [1,2] => [2,1] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,1,3] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [3,1,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1,2,4] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,2,4,3] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [4,2,1,3] => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [4,1,3,2] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [4,3,1,2] => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1,2,3,5] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,1,2,4] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [3,4,1,2,5] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,3,1,2,5] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,4,5,1,3] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,1,3] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [2,3,5,1,4] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [2,3,4,1,5] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,2,4,1,5] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,5,3,1,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,1,5] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [4,2,3,1,5] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,3,5,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,5,1,4,2] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,4,1,5,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,3,1,5,2] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,2,4,5,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,2,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [5,1,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [5,3,1,2,4] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,5,4,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [5,2,4,1,3] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [5,2,3,1,4] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [5,3,2,1,4] => 4
Description
The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000541
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ? = 1 - 1
[1,0,1,0]
 => [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1,2,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,2,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [4,1,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [4,2,1,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [4,1,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [4,3,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1,2,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,1,2,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [3,4,1,2,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,3,1,2,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,4,5,1,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,1,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [2,3,5,1,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [2,3,4,1,5] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,2,4,1,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,5,3,1,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,1,5] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [4,2,3,1,5] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,3,5,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,5,1,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,4,1,5,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,3,1,5,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,2,5,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [5,1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [5,3,1,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,5,4,1,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [5,2,4,1,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [5,2,3,1,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [5,3,2,1,4] => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,4,5,3,2] => 0 = 1 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000007
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [2,1] => [1,2] => 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [1,2,3] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,2,1,3] => [1,3,4,2] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,4,2,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,3,2,1,4] => [1,3,4,5,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,2,1,5,3] => [2,4,5,1,3] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,2,1,4,3] => [1,4,5,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,2,1,3,4] => [1,4,5,3,2] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,1,5,4,2] => [3,5,1,2,4] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,2,4] => [3,5,1,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1,5,3,2] => [2,5,1,3,4] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,1,4,3,2] => [1,5,2,3,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,1,4,2,3] => [1,5,2,4,3] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,1,3,5,2] => [2,5,3,1,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,4,2] => [1,5,3,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,1,3,2,4] => [1,5,3,4,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [2,5,3,1,4] => [4,1,3,5,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [2,5,1,4,3] => [4,1,5,2,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => [4,1,5,3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,5,4,2,1] => [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,5,2,1,4] => [3,1,4,5,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,5,3,2,1] => [2,1,3,4,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,2,1,3,5] => [2,4,5,3,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,4,5,2] => [3,5,2,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,4,2,5] => [3,5,2,4,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,1,3,2,5] => [2,5,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 4
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [2,3,7,6,5,4,1] => [6,5,1,2,3,4,7] => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => [2,4,7,6,5,3,1] => [6,4,1,2,3,5,7] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => [2,5,7,6,4,3,1] => [6,3,1,2,4,5,7] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,1,2,6] => [2,6,7,5,4,3,1] => [6,2,1,3,4,5,7] => ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,1,3,4] => [3,4,7,6,5,2,1] => [5,4,1,2,3,6,7] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [7,6,4,2,1,3,5] => [3,5,7,6,4,2,1] => [5,3,1,2,4,6,7] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [7,5,4,2,1,3,6] => [3,6,7,5,4,2,1] => [5,2,1,3,4,6,7] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [7,6,3,2,1,4,5] => [4,5,7,6,3,2,1] => [4,3,1,2,5,6,7] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [7,5,3,2,1,4,6] => [4,6,7,5,3,2,1] => [4,2,1,3,5,6,7] => ? = 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => [2,3,6,7,5,4,1] => [6,5,2,1,3,4,7] => ? = 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [2,4,5,7,6,3,1] => [6,4,3,1,2,5,7] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1,2,4,6] => [2,4,6,7,5,3,1] => [6,4,2,1,3,5,7] => ? = 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [2,5,6,7,4,3,1] => [6,3,2,1,4,5,7] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1,3,4,5] => [3,4,5,7,6,2,1] => [5,4,3,1,2,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,3,4,6] => [3,4,6,7,5,2,1] => [5,4,2,1,3,6,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => [3,5,6,7,4,2,1] => [5,3,2,1,4,6,7] => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [2,4,5,6,7,3,1] => [6,4,3,2,1,5,7] => ? = 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000991
(load all 5 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,2,1,3] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,1,3,2] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [2,4,3,1] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,4,2,1] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,3,2,1,4] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,2,1,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,2,1,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,2,1,3,4] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,1,5,4,2] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1,5,3,2] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,1,4,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,1,4,2,3] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,1,3,5,2] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,4,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,1,3,2,4] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [2,5,4,3,1] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [2,5,3,1,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [2,4,1,5,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,5,4,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,5,2,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,5,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,2,1,3,5] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,4,5,2] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,4,2,5] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [2,7,6,5,4,3,1] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => [3,7,6,5,4,2,1] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1,4] => [4,7,6,5,3,2,1] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1,5] => [5,7,6,4,3,2,1] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1,6] => [6,7,5,4,3,2,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [2,3,7,6,5,4,1] => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => [2,4,7,6,5,3,1] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => [2,5,7,6,4,3,1] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,1,2,6] => [2,6,7,5,4,3,1] => ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,1,3,4] => [3,4,7,6,5,2,1] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [7,6,4,2,1,3,5] => [3,5,7,6,4,2,1] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [7,5,4,2,1,3,6] => [3,6,7,5,4,2,1] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [7,6,3,2,1,4,5] => [4,5,7,6,3,2,1] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [7,5,3,2,1,4,6] => [4,6,7,5,3,2,1] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [7,4,3,2,1,5,6] => [5,6,7,4,3,2,1] => ? = 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => [2,3,6,7,5,4,1] => ? = 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [2,4,5,7,6,3,1] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1,2,4,6] => [2,4,6,7,5,3,1] => ? = 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [2,5,6,7,4,3,1] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1,3,4,5] => [3,4,5,7,6,2,1] => ? = 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,3,4,6] => [3,4,6,7,5,2,1] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => [3,5,6,7,4,2,1] => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => [4,5,6,7,3,2,1] => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [2,4,5,6,7,3,1] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,4,5,6,7,2,1] => ? = 1
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000314
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,2,1,3] => [3,2,4,1] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,1,3,2] => [2,4,3,1] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [2,4,3,1] => [4,1,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,4,2,1] => [4,3,1,2] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,2,1,5,3] => [3,2,5,1,4] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,2,1,4,3] => [3,2,5,4,1] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,2,1,3,4] => [3,2,4,5,1] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,1,5,4,2] => [2,5,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,2,4] => [2,4,1,5,3] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1,5,3,2] => [2,5,4,1,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,1,4,3,2] => [2,5,4,3,1] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,1,4,2,3] => [2,4,5,3,1] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,1,3,5,2] => [2,5,3,1,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,4,2] => [2,5,3,4,1] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,1,3,2,4] => [2,4,3,5,1] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [2,5,4,3,1] => [5,1,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [2,5,3,1,4] => [4,1,3,5,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [2,4,1,5,3] => [3,1,5,2,4] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [2,5,1,4,3] => [3,1,5,4,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => [3,1,4,5,2] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,5,4,2,1] => [5,4,1,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,5,2,1,4] => [4,3,1,5,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,5,3,2,1] => [5,4,3,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,2,1,3,5] => [3,2,4,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,4,5,2] => [2,5,1,3,4] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,4,2,5] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,1,3,2,5] => [2,4,3,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [2,7,6,5,4,3,1] => [7,1,6,5,4,3,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => [3,7,6,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1,4] => [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1,5] => [5,7,6,4,3,2,1] => [7,6,5,4,1,3,2] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1,6] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => [2,4,7,6,5,3,1] => [7,1,6,2,5,4,3] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => [2,5,7,6,4,3,1] => [7,1,6,5,2,4,3] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,1,2,6] => [2,6,7,5,4,3,1] => [7,1,6,5,4,2,3] => ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,1,3,4] => [3,4,7,6,5,2,1] => [7,6,1,2,5,4,3] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [7,6,4,2,1,3,5] => [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [7,5,4,2,1,3,6] => [3,6,7,5,4,2,1] => [7,6,1,5,4,2,3] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [7,6,3,2,1,4,5] => [4,5,7,6,3,2,1] => [7,6,5,1,2,4,3] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [7,5,3,2,1,4,6] => [4,6,7,5,3,2,1] => [7,6,5,1,4,2,3] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [7,4,3,2,1,5,6] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => [2,3,6,7,5,4,1] => [7,1,2,6,5,3,4] => ? = 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [2,4,5,7,6,3,1] => [7,1,6,2,3,5,4] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1,2,4,6] => [2,4,6,7,5,3,1] => [7,1,6,2,5,3,4] => ? = 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [2,5,6,7,4,3,1] => [7,1,6,5,2,3,4] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1,3,4,5] => [3,4,5,7,6,2,1] => [7,6,1,2,3,5,4] => ? = 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,3,4,6] => [3,4,6,7,5,2,1] => [7,6,1,2,5,3,4] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => [3,5,6,7,4,2,1] => [7,6,1,5,2,3,4] => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 1
Description
The number of left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$. This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000877
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => 10 => 01 => 01 => 1
[1,0,1,0]
 => 1010 => 0101 => 1001 => 1
[1,1,0,0]
 => 1100 => 0011 => 0011 => 2
[1,0,1,0,1,0]
 => 101010 => 010101 => 011001 => 1
[1,0,1,1,0,0]
 => 101100 => 010011 => 010011 => 2
[1,1,0,0,1,0]
 => 110010 => 001101 => 110001 => 1
[1,1,0,1,0,0]
 => 110100 => 001011 => 100011 => 2
[1,1,1,0,0,0]
 => 111000 => 000111 => 000111 => 3
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 10011001 => 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01010011 => 10010011 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 01001101 => 00111001 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 01001011 => 00110011 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 01000111 => 00100111 => 3
[1,1,0,0,1,0,1,0]
 => 11001010 => 00110101 => 10110001 => 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 00110011 => 10100011 => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 00101101 => 01110001 => 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 00101011 => 01100011 => 2
[1,1,0,1,1,0,0,0]
 => 11011000 => 00100111 => 01000111 => 3
[1,1,1,0,0,0,1,0]
 => 11100010 => 00011101 => 11100001 => 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 00011011 => 11000011 => 2
[1,1,1,0,1,0,0,0]
 => 11101000 => 00010111 => 10000111 => 3
[1,1,1,1,0,0,0,0]
 => 11110000 => 00001111 => 00001111 => 4
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101010011 => ? => ? = 2
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101001101 => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101001011 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101000111 => ? => ? = 3
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0100110101 => ? => ? = 2
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0100110011 => ? => ? = 3
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0100101101 => ? => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0100101011 => ? => ? = 2
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0100100111 => ? => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0100011101 => ? => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0100011011 => ? => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0100010111 => ? => ? = 3
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0100001111 => ? => ? = 4
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0011010101 => ? => ? = 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0011010011 => ? => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0011001101 => ? => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0011001011 => ? => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0011000111 => ? => ? = 3
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0010110101 => ? => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0010110011 => ? => ? = 2
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0010101101 => ? => ? = 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0010101011 => ? => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0010100111 => ? => ? = 3
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0010011101 => ? => ? = 2
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0010011011 => ? => ? = 3
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0010010111 => ? => ? = 3
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0010001111 => ? => ? = 4
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0001110101 => ? => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0001110011 => ? => ? = 2
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001101101 => ? => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0001101011 => ? => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0001100111 => ? => ? = 3
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0001011101 => ? => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 0001011011 => ? => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0001010111 => ? => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0001001111 => ? => ? = 4
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0000111101 => ? => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0000111011 => ? => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0000110111 => ? => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0000101111 => ? => ? = 4
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000011111 => ? => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 010101010101 => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => 010101010011 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 010101001101 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => 010101001011 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 010101000111 => ? => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => 010100110101 => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => 010100110011 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => 010100101101 => ? => ? = 2
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].