Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001733
(load all 11 compositions to match this statistic)
Mapping
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
Description
The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.
Matching statistic: St000541
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [3,1,2] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 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] => [2,3,1,4] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[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] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [5,4,3,1,2] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [5,4,1,3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [5,4,2,1,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [5,4,1,2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [5,1,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => [5,2,4,1,3] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [5,2,1,4,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => [5,3,2,1,4] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [5,3,1,2,4] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [5,1,2,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [5,1,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [5,2,1,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => [3,5,2,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [3,5,1,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => [3,2,5,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[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] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => [4,3,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [4,1,3,2,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => [4,2,1,3,5] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
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: St000542
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[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] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [5,4,3,1,2] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [5,4,1,3,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [5,4,2,1,3] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [5,4,1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [5,1,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => [5,2,4,1,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [5,2,1,4,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => [5,3,2,1,4] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [5,3,1,2,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [5,1,2,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [5,1,3,2,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [5,2,1,3,4] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [2,5,4,1,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => [3,5,2,1,4] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [3,5,1,2,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => [3,2,5,1,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 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] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => [4,3,1,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => [3,1,2,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [4,1,3,2,5] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => [4,2,1,3,5] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 0 + 1
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: St000068
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => ([],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => ([],3)
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [2,5,3,6,4,1] => ([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => [4,5,1,6,3,2] => ([(0,4),(1,2),(1,3),(1,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,1,2] => [3,4,1,6,5,2] => ([(0,3),(1,2),(1,4),(1,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => [3,5,4,1,6,2] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,1,2] => [4,3,5,1,6,2] => ([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [1,6,2,7,5,4,3] => ([(0,4),(0,5),(4,6),(5,1),(5,2),(5,3),(5,6)],7)
 => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,1,2,3] => [1,5,2,7,6,4,3] => ([(0,3),(0,4),(3,5),(3,6),(4,1),(4,2),(4,5),(4,6)],7)
 => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(2,6),(3,1),(3,4),(3,5),(3,6)],7)
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,1,2,4] => [1,6,3,2,7,5,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,1,2,4] => [1,5,3,2,7,6,4] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,7,5,1,2,3,4] => [1,2,6,3,7,5,4] => ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
 => [6,5,7,1,2,3,4] => [1,2,6,5,3,7,4] => ([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [6,7,4,1,2,3,5] => [1,2,6,4,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [7,4,5,1,2,3,6] => [1,4,2,5,3,7,6] => ([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [6,7,3,1,2,4,5] => [1,3,2,6,4,7,5] => ([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000061
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000061: Binary trees ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => [[.,.],.]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [.,[.,.]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [.,[[.,.],.]]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => [[[.,[.,.]],.],[.,.]]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
 => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [1,6,2,7,5,4,3] => [.,[[.,[[[.,.],.],.]],[.,.]]]
 => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,1,2,3] => [1,5,2,7,6,4,3] => [.,[[.,[[.,.],.]],[[.,.],.]]]
 => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => [.,[[.,[.,.]],[[[.,.],.],.]]]
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => [1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,1,2,4] => [1,6,3,2,7,5,4] => [.,[[[.,.],[[.,.],.]],[.,.]]]
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,1,2,4] => [1,5,3,2,7,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => [1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,7,5,1,2,3,4] => [1,2,6,3,7,5,4] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
 => [6,5,7,1,2,3,4] => [1,2,6,5,3,7,4] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [7,6,4,1,2,3,5] => [1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [6,7,4,1,2,3,5] => [1,2,6,4,3,7,5] => [.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => [1,2,6,5,4,3,7] => [.,[.,[[[[.,.],.],.],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [5,6,4,1,2,3,7] => [1,5,2,6,4,3,7] => [.,[[.,[[.,.],.]],[.,[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [7,4,5,1,2,3,6] => [1,4,2,5,3,7,6] => [.,[[.,[.,.]],[.,[[.,.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [6,4,5,1,2,3,7] => [1,4,2,6,5,3,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [5,4,6,1,2,3,7] => [1,5,4,2,6,3,7] => [.,[[[.,[.,.]],.],[.,[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [1,3,2,4,7,6,5] => [.,[[.,.],[.,[[[.,.],.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [6,7,3,1,2,4,5] => [1,3,2,6,4,7,5] => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1,2,4,6] => [1,3,2,5,4,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3,1,2,4,7] => [1,3,2,6,5,4,7] => [.,[[.,.],[[[.,.],.],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [5,6,3,1,2,4,7] => [1,5,3,2,6,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [1,4,3,2,5,7,6] => [.,[[[.,.],.],[.,[[.,.],.]]]]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3,1,2,5,7] => [1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => [1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => [1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [5,6,1,2,3,4,7] => [1,2,5,3,6,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [6,4,1,2,3,5,7] => [1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [4,5,1,2,3,6,7] => [1,4,2,5,3,6,7] => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,3,2,4,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,3,1,2,4,6,7] => [1,3,2,5,4,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => [1,4,3,2,5,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
 => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => ? = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ? = 0 + 1
Description
The number of nodes on the left branch of a binary tree. Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path.
Matching statistic: St000314
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,3,1,4] => [1,3,2,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,4,3] => [4,1,2,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,4,1] => [1,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,4,1] => [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,4,3,1] => [4,1,3,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,3,2,4,5] => [3,1,2,4,5] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,3,1,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [2,1,4,3,5] => [1,4,2,3,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,3,4,2,5] => [2,4,1,3,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [2,3,4,1,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,2,4,1,5] => [2,1,4,3,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,4,3,2,5] => [4,3,1,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,1,5] => [4,1,3,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,4,2,1,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [2,1,3,5,4] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,3,2,5,4] => [2,5,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [2,3,1,5,4] => [1,2,5,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,2,1,5,4] => [2,1,5,3,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,2,4,5,3] => [3,5,1,2,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [2,1,4,5,3] => [1,3,5,2,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,3,4,5,2] => [2,3,5,1,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [2,3,4,5,1] => [1,2,3,5,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,2,4,5,1] => [2,1,3,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,3,5,2] => [3,2,5,1,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,5,1] => [3,1,2,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,4,2,5,1] => [1,3,2,5,4] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => [3,2,1,5,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [2,1,3,4,7,6,5] => [7,1,6,2,3,4,5] => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,1,2,3] => [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,1,2,4] => [2,1,3,5,7,6,4] => [7,1,4,6,2,3,5] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,1,2,4] => [1,3,2,5,7,6,4] => [7,2,4,6,1,3,5] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,7,5,1,2,3,4] => [2,1,3,7,6,5,4] => [7,6,1,5,2,3,4] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [7,5,6,1,2,3,4] => [1,3,2,7,6,5,4] => [7,6,2,5,1,3,4] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
 => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => [7,6,1,2,5,3,4] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => [7,1,6,2,5,3,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [7,6,4,1,2,3,5] => [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [6,7,4,1,2,3,5] => [2,1,4,7,6,5,3] => [7,6,1,3,5,2,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => [7,6,2,3,5,1,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => [2,3,4,7,6,5,1] => [7,6,1,2,3,5,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [5,6,4,1,2,3,7] => [3,2,4,7,6,5,1] => [7,1,6,2,3,5,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [7,4,5,1,2,3,6] => [1,4,3,7,6,5,2] => [7,2,6,3,5,1,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [6,4,5,1,2,3,7] => [2,4,3,7,6,5,1] => [7,2,6,1,3,5,4] => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [5,4,6,1,2,3,7] => [3,4,2,7,6,5,1] => [7,1,2,6,3,5,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [1,2,5,7,6,4,3] => [7,3,6,5,1,2,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [6,7,3,1,2,4,5] => [2,1,5,7,6,4,3] => [7,3,6,1,5,2,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => [7,3,6,2,5,1,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3,1,2,4,7] => [2,3,5,7,6,4,1] => [7,3,6,1,2,5,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [5,6,3,1,2,4,7] => [3,2,5,7,6,4,1] => [7,1,3,6,2,5,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => [7,2,3,6,5,1,4] => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3,1,2,5,7] => [2,4,5,7,6,3,1] => [7,2,3,6,1,5,4] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => [7,6,5,1,4,2,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => [1,3,7,6,5,4,2] => [7,6,5,2,4,1,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => [2,3,7,6,5,4,1] => [7,6,5,1,2,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [5,6,1,2,3,4,7] => [3,2,7,6,5,4,1] => [7,6,1,5,2,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => [1,4,7,6,5,3,2] => [7,6,2,5,4,1,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [6,4,1,2,3,5,7] => [2,4,7,6,5,3,1] => [7,6,2,5,1,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => [3,4,7,6,5,2,1] => [7,6,1,2,5,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [4,5,1,2,3,6,7] => [4,3,7,6,5,2,1] => [7,1,6,2,5,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,5,7,6,4,3,2] => [7,2,6,5,4,1,3] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => [2,5,7,6,4,3,1] => [7,2,6,5,1,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,3,1,2,4,6,7] => [3,5,7,6,4,2,1] => [7,2,6,1,5,4,3] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => [4,5,7,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => [7,6,5,4,1,3,2] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => [7,6,5,1,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 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: St000991
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [2,3,4,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [4,1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[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] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [1,2,3,5,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [1,2,4,5,3] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [1,2,5,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [1,2,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => [1,3,5,2,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [1,4,5,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => [1,5,2,3,4] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [1,5,2,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [1,4,5,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [1,5,3,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [2,3,5,1,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => [2,4,5,1,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => [2,5,1,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => [3,4,5,1,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => [3,5,1,2,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => [4,5,1,2,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => [5,1,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => [4,5,1,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [5,1,3,4,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => [5,1,4,2,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [5,1,4,3,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [3,4,5,2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [1,6,2,7,5,4,3] => [3,4,5,7,2,6,1] => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,1,2,3] => [1,5,2,7,6,4,3] => [3,4,6,7,2,5,1] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => [3,5,6,7,2,4,1] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => [1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,1,2,4] => [1,6,3,2,7,5,4] => [4,5,7,2,3,6,1] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,1,2,4] => [1,5,3,2,7,6,4] => [4,6,7,2,3,5,1] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => [1,4,3,2,7,6,5] => [5,6,7,2,3,4,1] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,7,5,1,2,3,4] => [1,2,6,3,7,5,4] => [4,5,7,3,6,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
 => [6,5,7,1,2,3,4] => [1,2,6,5,3,7,4] => [4,7,3,5,6,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => [4,7,3,6,2,5,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [7,6,4,1,2,3,5] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [6,7,4,1,2,3,5] => [1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [5,6,4,1,2,3,7] => [1,5,2,6,4,3,7] => [7,3,4,6,2,5,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [7,4,5,1,2,3,6] => [1,4,2,5,3,7,6] => [6,7,3,5,2,4,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [6,4,5,1,2,3,7] => [1,4,2,6,5,3,7] => [7,3,5,6,2,4,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [5,4,6,1,2,3,7] => [1,5,4,2,6,3,7] => [7,3,6,2,4,5,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [6,7,3,1,2,4,5] => [1,3,2,6,4,7,5] => [5,7,4,6,2,3,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1,2,4,6] => [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3,1,2,4,7] => [1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [5,6,3,1,2,4,7] => [1,5,3,2,6,4,7] => [7,4,6,2,3,5,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [1,4,3,2,5,7,6] => [6,7,5,2,3,4,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3,1,2,5,7] => [1,4,3,2,6,5,7] => [7,5,6,2,3,4,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [5,6,1,2,3,4,7] => [1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [6,4,1,2,3,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [4,5,1,2,3,6,7] => [1,4,2,5,3,6,7] => [7,6,3,5,2,4,1] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,3,1,2,4,6,7] => [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => ? = 0 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => [1,4,3,2,5,6,7] => [7,6,5,2,3,4,1] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St001232
(load all 3 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 0
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 0
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 0
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 0
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.