Identifier
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0] => [3,1,2] => [2,3,1] => [2,3,1] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 3
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Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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.