Identifier
Values
[1] => [1] => [1] => [-1] => 0
[1,2] => [1,2] => [1,2] => [-1,-2] => 0
[2,1] => [2,1] => [2,1] => [-2,-1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
[2,3,1] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0
[3,1,2] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[3,2,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => 0
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[1,4,3,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => 0
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => 0
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 0
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0
[3,4,1,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
[4,1,3,2] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0
[4,2,1,3] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 0
[4,3,1,2] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,3,2,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => 0
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => 0
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => 0
[1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => 0
[1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => 0
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => 0
[1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => 0
[1,5,3,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0
[1,5,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0
[1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 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 number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
bar
Description
Return the signed permutation with all signs reversed.
Map
to signed permutation
Description
The signed permutation with all signs positive.