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] => 0
[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] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,0] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[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] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [1,3,4,2] => [1,3,4,2] => 0
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[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] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[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] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
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Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
  • $-i < -j < -\pi(j) < -\pi(i)$, or
  • $-i < j \leq \pi(j) < -\pi(i)$, or
  • $i < j \leq \pi(j) < \pi(i)$.
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
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].