Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000337: Permutations ⟶ ℤ
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
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,3,4,6,5] => 1
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 2
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,5,6,3,2,4] => 1
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,6,4,5,2,3] => 3
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,6,3,2,4,5] => 1
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,4,5,6,2,3] => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,2,6,4,3,5] => 1
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,2,5,6,3,4] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,6,4,2,3,5] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,2,3,6,4,5] => 2
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,6,3,4,2,5] => 2
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,5,6,2,3,4] => 3
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,2,6,3,4,5] => 3
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,6,2,3,4,5] => 4
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => 3
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [1,6,7,5,4,3,2] => 2
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => 2
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => 2
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => 2
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => 2
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,7,6,4,5,3,2] => 3
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,7,4,5,6,3,2] => 4
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => 2
[[[[[[]]]]],[]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [1,2,3,4,5,7,6] => 1
[[[[],[[[]]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => 3
[[[[[],[[]]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [1,6,7,2,3,4,5] => 4
[[[[[[],[]]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [1,7,2,3,4,5,6] => 5
[[[[[[[]]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[[],[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => 3
[[],[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [1,7,8,6,5,4,3,2] => 3
[[],[],[],[],[[]],[]] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [6,1,2,3,4,8,5,7] => [1,6,8,7,5,4,3,2] => 3
[[],[],[],[],[[],[]]] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [8,1,2,3,4,7,5,6] => [1,8,6,7,5,4,3,2] => 4
[[],[],[],[],[[[]]]] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,1,2,3,4,7,8,5] => [1,6,7,8,5,4,3,2] => 2
[[],[],[],[[]],[],[]] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [5,1,2,3,8,4,6,7] => [1,5,8,7,6,4,3,2] => 3
[[],[],[],[[],[]],[]] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [8,1,2,3,6,4,5,7] => [1,8,7,5,6,4,3,2] => 3
[[],[],[[[],[]]],[]] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [8,1,2,5,6,3,4,7] => [1,8,7,4,5,6,3,2] => 4
[[],[],[[],[[]],[]]] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [8,1,2,5,3,7,4,6] => [1,8,6,7,4,5,3,2] => 4
[[],[[]],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => [1,3,5,7,8,6,4,2] => 2
[[],[[],[]],[],[[]]] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [7,1,4,2,3,5,8,6] => [1,7,8,6,5,3,4,2] => 3
[[],[[[],[[[]]]]]] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [6,1,4,5,2,7,8,3] => [1,6,7,8,3,4,5,2] => 3
[[[]],[],[],[[[]]]] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,6,1,3,4,7,8,5] => [1,2,6,7,8,5,4,3] => 2
[[[]],[[[],[]]],[]] => [1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [2,8,1,5,6,3,4,7] => [1,2,8,7,4,5,6,3] => 3
[[[],[]],[],[],[],[]] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,3,1,2,4,5,6,7] => [1,8,7,6,5,4,2,3] => 4
[[[[]]],[[[[]]]]] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [2,3,5,1,6,7,8,4] => [1,2,3,5,6,7,8,4] => 1
[[[],[[]]],[[],[]]] => [1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [4,3,1,8,2,7,5,6] => [1,4,8,6,7,5,2,3] => 4
[[[[[]],[]]],[],[]] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [2,8,4,5,1,3,6,7] => [1,2,8,7,6,3,4,5] => 4
[[[[[[[]]]]]],[]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => [1,2,3,4,5,6,8,7] => 1
[[[[[[[]]]]],[]]] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [2,3,4,5,8,7,1,6] => [1,2,3,4,5,8,6,7] => 2
[[[[[],[[[]]]]]]] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [6,3,4,5,1,7,8,2] => [1,6,7,8,2,3,4,5] => 4
[[[[[[[]]],[]]]]] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [2,3,8,5,6,7,1,4] => [1,2,3,8,4,5,6,7] => 4
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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 $\tau_{i}$.
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 $\tau_{i}$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
This sends the maximal height of the Dyck path to the depth of the tree.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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