searching the database
Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,4,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,4,3,5,2,6] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,6,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,5,3,6,2,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,4,3,6,5,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,4,3,5,6,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => [1,4,3,2,6,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => [1,5,3,4,6,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => [1,4,3,2,5,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => [1,5,3,4,2,6] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => [1,4,5,2,3,6] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,3,5,4,2,6] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => [1,4,6,2,5,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => [1,5,6,4,2,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => [1,3,6,4,5,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => [1,4,5,2,6,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => [1,3,5,4,6,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => [1,3,4,2,6,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => [1,2,5,4,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => [1,3,4,2,5,6] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000374
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [7,2,1,4,3,6,5] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [4,2,6,1,5,3,7] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,6,1,7,3,4,5] => [6,2,7,3,4,1,5] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,7,4,5] => [6,1,3,7,4,2,5] => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5,7] => [3,1,2,6,4,5,7] => ? = 4
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000703
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 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]
=> [3,2,1,4,5] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 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]
=> [4,3,2,1,5,6] => 2
[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]
=> [3,4,2,1,5,6] => 2
[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]
=> [4,2,3,1,5,6] => 1
[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]
=> [3,2,4,1,5,6] => 1
[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]
=> [2,3,4,1,5,6] => 1
[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]
=> [4,3,1,2,5,6] => 2
[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]
=> [3,4,1,2,5,6] => 2
[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]
=> [4,2,1,3,5,6] => 2
[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]
=> [3,2,1,4,5,6] => 1
[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]
=> [2,3,1,4,5,6] => 1
[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]
=> [4,1,2,3,5,6] => 3
[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]
=> [3,1,2,4,5,6] => 2
[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]
=> [2,1,3,4,5,6] => 1
[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]
=> [1,2,3,4,5,6] => 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]
=> [5,4,3,2,1,6,7] => 2
[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]
=> [4,5,3,2,1,6,7] => ? = 2
[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]
=> [5,3,4,2,1,6,7] => ? = 2
[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]
=> [4,3,5,2,1,6,7] => ? = 2
[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]
=> [3,4,5,2,1,6,7] => ? = 2
[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]
=> [5,4,2,3,1,6,7] => ? = 3
[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]
=> [4,5,2,3,1,6,7] => ? = 3
[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]
=> [5,3,2,4,1,6,7] => ? = 2
[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]
=> [4,3,2,5,1,6,7] => ? = 2
[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]
=> [3,4,2,5,1,6,7] => ? = 2
[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]
=> [5,2,3,4,1,6,7] => ? = 1
[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]
=> [4,2,3,5,1,6,7] => ? = 1
[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]
=> [3,2,4,5,1,6,7] => ? = 1
[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]
=> [2,3,4,5,1,6,7] => 1
[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]
=> [5,4,3,1,2,6,7] => ? = 2
[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]
=> [4,5,3,1,2,6,7] => ? = 2
[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]
=> [5,3,4,1,2,6,7] => ? = 2
[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]
=> [4,3,5,1,2,6,7] => ? = 2
[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]
=> [3,4,5,1,2,6,7] => ? = 2
[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]
=> [5,4,2,1,3,6,7] => ? = 3
[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]
=> [4,5,2,1,3,6,7] => ? = 3
[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]
=> [5,3,2,1,4,6,7] => ? = 3
[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]
=> [4,3,2,1,5,6,7] => 2
[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]
=> [3,4,2,1,5,6,7] => ? = 2
[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]
=> [5,2,3,1,4,6,7] => ? = 2
[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]
=> [4,2,3,1,5,6,7] => ? = 1
[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]
=> [3,2,4,1,5,6,7] => ? = 1
[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]
=> [2,3,4,1,5,6,7] => 1
[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]
=> [5,4,1,2,3,6,7] => ? = 2
[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]
=> [4,5,1,2,3,6,7] => ? = 3
[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]
=> [5,3,1,2,4,6,7] => ? = 3
[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]
=> [4,3,1,2,5,6,7] => ? = 2
[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]
=> [3,4,1,2,5,6,7] => 2
[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]
=> [5,2,1,3,4,6,7] => ? = 3
[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]
=> [4,2,1,3,5,6,7] => ? = 2
[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]
=> [3,2,1,4,5,6,7] => 1
[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]
=> [2,3,1,4,5,6,7] => 1
[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]
=> [5,1,2,3,4,6,7] => 4
[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]
=> [4,1,2,3,5,6,7] => 3
[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]
=> [3,1,2,4,5,6,7] => 2
[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]
=> [2,1,3,4,5,6,7] => 1
[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]
=> [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,5,4,3,2,1,7,8] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,1,7,8] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,2,1,7,8] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,2,1,7,8] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,2,1,7,8] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,2,1,7,8] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,1,7,8] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,1,7,8] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,2,1,7,8] => ? = 2
[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]
=> [3,4,5,6,2,1,7,8] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,2,3,1,7,8] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,1,7,8] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,3,1,7,8] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [6,5,3,2,4,1,7,8] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,2,4,1,7,8] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [6,4,3,2,5,1,7,8] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,6,1,7,8] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,5,1,7,8] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,3,4,2,6,1,7,8] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,3,4,1,7,8] => ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,1,2,7,8] => 2
[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]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1,6,7,8] => 2
[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]
=> [2,3,4,5,1,6,7,8] => 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,1,2,3,7,8] => 3
[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]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 2
[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]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [3,4,1,2,5,6,7,8] => 2
[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]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8] => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [6,1,2,3,4,5,7,8] => 5
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => 4
[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]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => 1
[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]
=> [1,2,3,4,5,6,7,8] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,6,5,4,3,2,1,8,9] => 3
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => 1
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St001769
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0
[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]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 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]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 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]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,2,4,1,6,5] => [3,2,4,1,6,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,4,6,3] => [2,1,5,4,6,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [3,2,5,4,6,1] => [3,2,5,4,6,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,2,4,1,5,6] => [3,2,4,1,5,6] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [3,2,5,4,1,6] => [3,2,5,4,1,6] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,4,5,3] => [2,1,6,4,5,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [3,2,6,4,5,1] => [3,2,6,4,5,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,5,6,3,4] => [2,1,5,6,3,4] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,2,5,6,1,4] => [3,2,5,6,1,4] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,4,6,5,3] => [2,1,4,6,5,3] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [3,2,4,6,5,1] => [3,2,4,6,5,1] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,4,5,3,6] => [2,1,4,5,3,6] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,6] => [3,2,4,5,1,6] => [3,2,4,5,1,6] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => [2,1,3,6,5,4] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,4,5,6,3] => [2,1,4,5,6,3] => ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [3,2,4,5,6,1] => [3,2,4,5,6,1] => ? = 4
Description
The reflection length of a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
Matching statistic: St001864
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0
[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]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 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]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 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]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,4,2,5,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,1,4,2,6,5] => [3,1,4,2,6,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,3,6,4] => [2,1,5,3,6,4] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,1,5,2,6,3] => [4,1,5,2,6,3] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,1,5,2,3,6] => [4,1,5,2,3,6] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,3,4,5] => [2,1,6,3,4,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [5,1,6,2,3,4] => [5,1,6,2,3,4] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,4,6,3,5] => [2,1,4,6,3,5] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,1,4,6,2,5] => [3,1,4,6,2,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,5,6,3,4] => [2,1,5,6,3,4] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [4,1,5,6,2,3] => [4,1,5,6,2,3] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,4,5,3,6] => [2,1,4,5,3,6] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,6] => [3,1,4,5,2,6] => [3,1,4,5,2,6] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,4,5] => [2,1,3,6,4,5] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,4,5,6,3] => [2,1,4,5,6,3] => ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? = 4
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: St000120
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 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
[1,0,1,1,0,0]
=> [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]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[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
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0]
=> [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
[1,0,1,0,1,1,0,0,1,0]
=> [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
[1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 3
[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
[1,0,1,1,0,1,0,0,1,0]
=> [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
[1,0,1,1,0,1,0,1,0,0]
=> [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
[1,0,1,1,0,1,1,0,0,0]
=> [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
[1,0,1,1,1,0,0,0,1,0]
=> [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
[1,0,1,1,1,0,0,1,0,0]
=> [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
[1,0,1,1,1,0,1,0,0,0]
=> [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
[1,0,1,1,1,1,0,0,0,0]
=> [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
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> ? = 2
[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
[1,1,0,0,1,1,0,0,1,0]
=> [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
[1,1,0,0,1,1,0,1,0,0]
=> [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
[1,1,0,0,1,1,1,0,0,0]
=> [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
[1,1,0,1,0,0,1,0,1,0]
=> [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
[1,1,0,1,0,0,1,1,0,0]
=> [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
[1,1,0,1,0,1,0,0,1,0]
=> [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]
=> ? = 3
[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
[1,1,0,1,0,1,1,0,0,0]
=> [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
[1,1,0,1,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,0,0,1,0,0]
=> [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]
=> ? = 1
[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
[1,1,0,1,1,1,0,0,0,0]
=> [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
[1,1,1,0,0,0,1,0,1,0]
=> [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]
=> ? = 2
[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
[1,1,1,0,0,1,0,0,1,0]
=> [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
[1,1,1,0,0,1,0,1,0,0]
=> [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
[1,1,1,0,0,1,1,0,0,0]
=> [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
[1,1,1,0,1,0,0,0,1,0]
=> [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
[1,1,1,0,1,0,0,1,0,0]
=> [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]
=> ? = 2
[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
[1,1,1,0,1,1,0,0,0,0]
=> [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
[1,1,1,1,0,0,0,0,1,0]
=> [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
[1,1,1,1,0,0,0,1,0,0]
=> [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
[1,1,1,1,0,0,1,0,0,0]
=> [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
[1,1,1,1,0,1,0,0,0,0]
=> [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
[1,1,1,1,1,0,0,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 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: St000316
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 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]
=> [1,2,4,3,5] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 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]
=> [1,2,4,3,6,5] => 2
[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]
=> [1,2,4,6,3,5] => 2
[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]
=> [1,2,4,3,5,6] => 1
[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]
=> [1,2,4,5,3,6] => 1
[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]
=> [1,2,4,5,6,3] => 1
[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]
=> [1,2,5,3,6,4] => 2
[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]
=> [1,2,5,6,3,4] => 2
[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]
=> [1,2,5,3,4,6] => 2
[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]
=> [1,2,3,5,4,6] => 1
[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]
=> [1,2,3,5,6,4] => 1
[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]
=> [1,2,6,3,4,5] => 3
[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]
=> [1,2,3,6,4,5] => 2
[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]
=> [1,2,3,4,6,5] => 1
[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]
=> [1,2,3,4,5,6] => 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]
=> [1,2,4,3,6,5,7] => ? = 2
[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]
=> [1,2,4,6,3,5,7] => ? = 2
[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]
=> [1,2,4,3,6,7,5] => ? = 2
[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]
=> [1,2,4,6,3,7,5] => ? = 2
[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]
=> [1,2,4,6,7,3,5] => ? = 2
[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]
=> [1,2,4,3,7,5,6] => ? = 3
[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]
=> [1,2,4,7,3,5,6] => ? = 3
[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]
=> [1,2,4,3,5,7,6] => ? = 2
[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]
=> [1,2,4,5,3,7,6] => ? = 2
[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]
=> [1,2,4,5,7,3,6] => ? = 2
[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]
=> [1,2,4,3,5,6,7] => ? = 1
[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]
=> [1,2,4,5,3,6,7] => ? = 1
[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]
=> [1,2,4,5,6,3,7] => ? = 1
[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]
=> [1,2,4,5,6,7,3] => ? = 1
[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]
=> [1,2,5,3,6,4,7] => ? = 2
[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]
=> [1,2,5,6,3,4,7] => ? = 2
[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]
=> [1,2,5,3,6,7,4] => ? = 2
[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]
=> [1,2,5,6,3,7,4] => ? = 2
[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]
=> [1,2,5,6,7,3,4] => ? = 2
[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]
=> [1,2,5,3,7,4,6] => ? = 3
[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]
=> [1,2,5,7,3,4,6] => ? = 3
[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]
=> [1,2,5,3,4,7,6] => ? = 3
[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]
=> [1,2,3,5,4,7,6] => ? = 2
[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]
=> [1,2,3,5,7,4,6] => ? = 2
[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]
=> [1,2,5,3,4,6,7] => ? = 2
[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]
=> [1,2,3,5,4,6,7] => ? = 1
[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]
=> [1,2,3,5,6,4,7] => ? = 1
[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]
=> [1,2,3,5,6,7,4] => ? = 1
[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]
=> [1,2,6,3,7,4,5] => ? = 2
[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]
=> [1,2,6,7,3,4,5] => ? = 3
[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]
=> [1,2,6,3,4,7,5] => ? = 3
[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]
=> [1,2,3,6,4,7,5] => ? = 2
[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]
=> [1,2,3,6,7,4,5] => ? = 2
[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]
=> [1,2,6,3,4,5,7] => ? = 3
[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]
=> [1,2,3,6,4,5,7] => ? = 2
[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]
=> [1,2,3,4,6,5,7] => ? = 1
[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]
=> [1,2,3,4,6,7,5] => ? = 1
[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]
=> [1,2,7,3,4,5,6] => ? = 4
[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]
=> [1,2,3,7,4,5,6] => ? = 3
[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]
=> [1,2,3,4,7,5,6] => ? = 2
[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]
=> [1,2,3,4,5,7,6] => ? = 1
[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]
=> [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5,8,7] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,6,8,3,5,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5,7,8] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,2,4,3,6,7,5,8] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,2,4,6,7,3,5,8] => ? = 2
[]
=> [1,0]
=> [1,1,0,0]
=> [1,2] => 0
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St001905
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [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] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,1,2] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,2,1,3] => ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,3,2,1,4] => ? = 3
[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] => [3,4,2,1,5] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,2,3,1,4] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 3
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,4,5,3,2,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,3,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [5,6,3,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [6,4,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,5,3,6,2,1] => ? = 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 $a_i < i$.
Matching statistic: St001946
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0
[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] => [2,3,1] => [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [3,2,1] => 2
[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] => [3,2,4,1] => [3,2,4,1] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => [4,2,3,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2
[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] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 3
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [4,3,6,1,5,2] => [4,3,6,1,5,2] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [5,2,4,3,6,1] => [5,2,4,3,6,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,4,5,2,6,1] => [3,4,5,2,6,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [3,4,6,1,5,2] => [3,4,6,1,5,2] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [3,5,2,4,6,1] => [3,5,2,4,6,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [3,5,2,6,1,4] => [3,5,2,6,1,4] => ? = 2
Description
The number of descents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St001960
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,2,4] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,5,4,2,3,6] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,6,4,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,4,2,3,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,5,2,3,6,4] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,3,2,6,5,4] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,2,3] => ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => [1,4,2,3,6,5] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => [1,6,5,2,3,4] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => [1,6,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => [1,5,3,2,4,6] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,4,2,5,3,6] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => [1,6,3,2,4,5] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => [1,4,2,6,3,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => [1,6,3,5,2,4] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => [1,6,5,3,2,4] => ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => [1,4,2,6,5,3] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => [1,4,3,2,6,5] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => [1,2,6,5,3,4] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => [1,4,3,2,5,6] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => [1,2,6,3,4,5] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => [1,5,3,2,6,4] => ? = 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,6,2,3,4] => [1,6,4,2,5,3] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4] => [1,6,4,3,2,5] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,2,5,3,6,4] => [1,2,6,4,3,5] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4] => [1,2,5,3,6,4] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,2,3,4,6] => [1,5,4,3,2,6] => ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,5,3,4,6] => [1,2,5,4,3,6] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,4,5] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => [1,6,5,4,3,2] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [1,2,6,5,4,3] => ? = 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => [1,2,3,6,5,4] => ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,7,6] => [1,5,4,2,3,7,6] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,6] => [1,3,2,7,6,4,5] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6] => [1,5,2,3,7,6,4] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,6,7] => [1,5,4,2,3,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ? = 2
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!