Identifier
Values
[] => [] => [1] => [1] => 0
[[]] => [1,0] => [2,1] => [2,1] => 1
[[],[]] => [1,0,1,0] => [3,1,2] => [3,1,2] => 1
[[[]]] => [1,1,0,0] => [2,3,1] => [3,2,1] => 1
[[],[],[]] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[[],[[]]] => [1,0,1,1,0,0] => [3,1,4,2] => [4,3,1,2] => 1
[[[]],[]] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 1
[[[],[]]] => [1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => 2
[[[[]]]] => [1,1,1,0,0,0] => [2,3,4,1] => [4,3,2,1] => 2
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,4,1,2,3] => 1
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,3,1,2,4] => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,2,5,3] => 2
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,4,3,1,2] => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,4,2,1,3] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => 2
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,3,2,1,4] => 2
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,5,2,3] => 2
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,1,5,4,2] => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,3,1,5,2] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,4,3,2,1] => 2
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,5,1,2,3,4] => 1
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,4,1,2,3,5] => 1
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,2,3,6,4] => 2
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,5,4,1,2,3] => 2
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,3,1,2,4,5] => 1
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,5,3,1,2,4] => 2
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,2,6,3,5] => 2
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,4,3,1,2,5] => 2
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,1,2,6,3,4] => 2
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,2,6,5,3] => 2
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,3,1,2,6,4] => 2
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [5,4,1,2,6,3] => 2
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,5,4,3,1,2] => 2
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 1
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,5,2,1,3,4] => 2
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,4,2,1,3,5] => 2
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,1,3,6,4] => 2
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 2
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,3,2,1,4,5] => 2
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,6,5,2,4] => 2
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,5,3,2,1,4] => 2
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,6,2,3,5] => 2
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,1,6,4,2,5] => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 2
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,3,1,6,2,5] => 2
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,4,3,2,1,5] => 2
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,5,2,3,4] => 2
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,1,6,5,2,3] => 2
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,6,4] => 3
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,4,1,6,2,3] => 2
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,1,6,5,4,2] => 3
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,6,3,4] => 2
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,1,6,5,3] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,3,1,6,2,4] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,3,2,1,6,4] => 3
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,5,2,6,3] => 3
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,3,1,6,5,2] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,4,2,1,6,3] => 3
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,4,3,1,6,2] => 3
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,5,4,3,2,1] => 3
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Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
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.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.