Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000662
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => [3,1,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [3,2,1] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,4,2,1] => [4,1,3,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,4,1] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [2,3,4,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,1,3,2] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => [5,4,1,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => [4,2,5,1,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,4,2,5,1] => [5,1,3,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,4,1,3,5] => [4,3,1,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,1,3,4] => [5,4,3,1,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,5,2,4,1] => [4,5,1,3,2] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,4,2,1,5] => [4,1,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,5,2,1,4] => [5,4,1,3,2] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,5,3,2,1] => [3,5,1,4,2] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,5,1] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2,4,1,5] => [2,4,1,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [4,3,5,2,1] => [5,1,4,2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,2,3,5,1] => [2,3,5,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,2,3,1,4] => [2,3,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,3,4,2,1] => [4,2,3,5,1] => 2
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000120
Mapping
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 40%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? = 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 3
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 3
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 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]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 2
[]
 => []
 => [1,0]
 => [1,1,0,0]
 => 0
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St001864
(load all 3 compositions to match this statistic)
Mapping
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 40%
Values
[1,0]
 => [.,.]
 => [1] => [1] => 0
[1,0,1,0]
 => [[.,.],.]
 => [1,2] => [1,2] => 0
[1,1,0,0]
 => [.,[.,.]]
 => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,4,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [5,4,1,2,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [5,3,1,2,4] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [3,4,5,1,2] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,3,4,1,2] => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [4,3,5,1,2] => ? = 3
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,5,2,1,3] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,4,2,1,3] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,4,2,3,1] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[1,1,1,0,0,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3
[1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,4,2,1,5] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[1,1,1,1,0,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => ? = 3
[1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[[[[.,.],.],.],[.,.]],.]
 => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,.]],.],.]
 => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => [6,4,1,2,3,5] => [6,4,1,2,3,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[[[.,.],.],[[.,.],.]],.]
 => [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3
Description
The number of excedances of a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001905
(load all 4 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,4,2,1] => [3,4,2,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,3,4,5,1,6] => [2,3,4,5,1,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,4,6,1,5] => [2,3,4,6,1,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [3,4,5,6,2,1] => [3,4,5,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [2,3,4,1,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [3,4,5,2,6,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [2,3,5,1,4,6] => [2,3,5,1,4,6] => ? = 2
Description
The number of preferred parking spots in a parking function less than the index of the car. Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.