Identifier
Values
[] => [] => [1] => [1] => 0
[[]] => [1,0] => [2,1] => [2,1] => 0
[[],[]] => [1,0,1,0] => [3,1,2] => [2,3,1] => 0
[[[]]] => [1,1,0,0] => [2,3,1] => [3,1,2] => 0
[[],[],[]] => [1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => 0
[[],[[]]] => [1,0,1,1,0,0] => [3,1,4,2] => [4,2,3,1] => 1
[[[]],[]] => [1,1,0,0,1,0] => [2,4,1,3] => [3,4,1,2] => 0
[[[],[]]] => [1,1,0,1,0,0] => [4,3,1,2] => [2,4,3,1] => 1
[[[[]]]] => [1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 0
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,5,3,4,1] => 1
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,5,2,3,1] => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,4,2,5,1] => 1
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,3,1,4] => 1
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,4,5,1,2] => 0
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,4,1,2] => 1
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [2,4,5,3,1] => 1
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,5,1,2,3] => 0
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [2,3,5,4,1] => 1
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,4,3,1] => 1
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,5,4,1,2] => 1
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,4,1,5,3] => 1
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,6,4,5,1] => 1
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,5,6,3,4,1] => 1
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,4,5,3,6,1] => 1
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,6,3,4,1,5] => 1
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,5,6,2,3,1] => 1
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,5,2,3,1] => 1
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [3,4,2,5,6,1] => 1
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,6,2,3,1,4] => 1
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [3,4,5,2,6,1] => 1
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,5,1] => 1
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,6,5,2,3,1] => 1
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [3,5,2,4,6,1] => 2
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,3,1,4,5] => 1
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,4,5,6,1,2] => 0
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,4,5,1,2] => 1
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [5,6,3,4,1,2] => 1
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [4,5,3,6,1,2] => 1
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,4,1,2,5] => 1
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [2,4,5,6,3,1] => 1
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,5,6,1,2,3] => 0
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [2,6,4,5,3,1] => 2
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,5,1,2,3] => 1
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [2,3,5,6,4,1] => 1
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,6,2,4,3,1] => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [3,5,6,4,1,2] => 1
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => 1
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,6,1,2,3,4] => 0
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [2,3,4,6,1,5] => 1
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [2,6,3,5,4,1] => 1
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,5,2,6,3,1] => 1
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,2,6,4,1] => 2
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,2,4,3,1,5] => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [3,4,6,5,1,2] => 1
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [6,3,5,4,1,2] => 1
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [2,4,5,1,6,3] => 1
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [4,6,5,1,2,3] => 1
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [2,3,6,5,4,1] => 1
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,2,4,1,5,3] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,5,1,2,6,4] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [2,5,1,4,6,3] => 1
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
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Description
The cyclic holeyness of a permutation.
For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $(m\bmod n)+1\notin S$.
For a permutation $\pi$ of $[n]$ the cyclic holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$
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
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.