Identifier
Values
[1,0] => [1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [1,0,1,0] => [3,1,2] => [2,3,1] => 1
[1,1,0,0] => [1,1,0,0] => [2,3,1] => [3,1,2] => 1
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => [2,4,3,1] => 2
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => [3,4,1,2] => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => [4,2,3,1] => 1
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [2,3,5,4,1] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [2,4,5,3,1] => 2
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,4,1,5,3] => 2
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,4,5,1,2] => 2
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,5,4,1,2] => 2
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,5,2,3,1] => 2
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,5,3,4,1] => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,3,1,4] => 2
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,5,1,2,3] => 2
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,4,1,2] => 2
[1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,4,3,1] => 1
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 1
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Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
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
Adin-Bagno-Roichman transformation
Description
The Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and sending the number of returns to the number of up steps after the last double up step.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.