Your data matches 23 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000120
(load all 5 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
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
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [2,1] => 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
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: St000337
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [2,1] => 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000374
(load all 4 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [2,1] => 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 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
(load all 10 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => 0
[[],[]]
 => [.,[.,.]]
 => [2,1] => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => 0
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => 2
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => 0
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 2
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 2
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 3
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 2
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 2
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 2
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => 2
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 2
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => 3
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => 3
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => 2
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 2
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 2
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => 2
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => 2
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 2
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => 2
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 3
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 3
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 3
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 2
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 3
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 3
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 4
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: St000021
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => [1] => 0
[[],[]]
 => [.,[.,.]]
 => [2,1] => [2,1] => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => [1,2] => 0
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => [3,2,1] => 2
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,4,1] => 2
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => 2
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,4,1] => 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,4,1,3] => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,1,2,3] => 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,2,3,1] => 2
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,1,4,2] => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,3,1] => 2
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,3,2,1] => 3
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1,2,4] => 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,2,1,4] => 2
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [3,4,2,5,1] => 2
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [3,5,1,4,2] => 2
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [4,2,3,5,1] => 2
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,2,3] => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,2,5,1,3] => 2
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,3,2,5,1] => 3
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,1,4,3,2] => 3
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,5,1] => 2
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2,5,1,4] => 2
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,1,3,2,4] => 2
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,3,5,1,4] => 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,5,1,3,4] => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [3,5,2,4,1] => 2
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [3,4,1,5,2] => 2
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,2,3,4,1] => 2
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [4,1,5,2,3] => 2
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,2,4,1,3] => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,3,2,4,1] => 3
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,2,5,3,1] => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,1,5,3,2] => 3
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,3,1,4,2] => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,5,4,1] => 3
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,5,4,1] => 2
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [5,4,2,3,1] => 3
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,4,3,1] => 3
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,4,3,2,1] => 4
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000024
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000155
(load all 6 compositions to match this statistic)
Mapping
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [[]]
 => [.,.]
 => [1] => 0
[[],[]]
 => [[],[]]
 => [.,[.,.]]
 => [2,1] => 1
[[[]]]
 => [[[]]]
 => [[.,.],.]
 => [1,2] => 0
[[],[],[]]
 => [[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1
[[],[[]]]
 => [[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => 1
[[[]],[]]
 => [[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 2
[[[],[]]]
 => [[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => 1
[[[[]]]]
 => [[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => 0
[[],[],[],[]]
 => [[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 2
[[],[],[[]]]
 => [[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2
[[],[[]],[]]
 => [[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 1
[[],[[],[]]]
 => [[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1
[[],[[[]]]]
 => [[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 1
[[[]],[],[]]
 => [[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[[[]],[[]]]
 => [[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2
[[[],[]],[]]
 => [[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2
[[[[]]],[]]
 => [[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 3
[[[],[],[]]]
 => [[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1
[[[],[[]]]]
 => [[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1
[[[[]],[]]]
 => [[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2
[[[[],[]]]]
 => [[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1
[[[[[]]]]]
 => [[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 2
[[],[],[],[[]]]
 => [[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => 2
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => 2
[[],[],[[],[]]]
 => [[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 2
[[],[],[[[]]]]
 => [[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => 2
[[],[[]],[],[]]
 => [[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => 3
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 3
[[],[[],[]],[]]
 => [[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => 2
[[],[[[]]],[]]
 => [[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => 1
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 2
[[],[[],[[]]]]
 => [[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 2
[[],[[[]],[]]]
 => [[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 1
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 1
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 1
[[[]],[],[],[]]
 => [[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 2
[[[]],[],[[]]]
 => [[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => 2
[[[]],[[]],[]]
 => [[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => 2
[[[]],[[],[]]]
 => [[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 2
[[[]],[[[]]]]
 => [[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => 2
[[[],[]],[],[]]
 => [[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 3
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 3
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => 3
[[[[]]],[[]]]
 => [[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => 3
[[[],[],[]],[]]
 => [[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 3
[[[],[[]]],[]]
 => [[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => 2
[[[[]],[]],[]]
 => [[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 3
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 3
[[[[[]]]],[]]
 => [[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 4
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000329
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000996
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [2,1] => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,2] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,3,2] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [2,1,3] => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 2
[[],[[[]],[]]]
 => [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] => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
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.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000676The number of odd rises of a Dyck path. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000702The number of weak deficiencies of a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car.