Identifier
Values
[1,0] => [1,0] => [1] => [1] => 0
[1,0,1,0] => [1,0,1,0] => [1,2] => [1,2] => 0
[1,1,0,0] => [1,1,0,0] => [2,1] => [2,1] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => [2,3,1] => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [3,1,2] => [3,1,2] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => [3,4,1,2] => 0
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,1,4,2] => 0
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [2,4,1,3] => [2,4,1,3] => 0
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,4,2] => 0
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4,2,3,5] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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
Barnabei-Castronuovo involution
Description
The Barnabei-Castronuovo Schützenberger involution on Dyck paths.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let $D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is $\dots, 1, D_2, 1, D_1$.
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.