Your data matches 1 statistic following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000337
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => []
 => [1] => [1] => 0
[[]]
 => [1,0]
 => [2,1] => [1,2] => 0
[[],[]]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 1
[[[]]]
 => [1,1,0,0]
 => [2,3,1] => [1,2,3] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => 3
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => 3
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [1,2,3,6,5,4] => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [1,2,3,5,6,4] => 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [1,2,6,5,3,4] => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [1,6,5,2,3,4] => 3
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 $\tau_{i}$.