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] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 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] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 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] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,3,1] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[3,1,2,4] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,3,2] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,3,1,2] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 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] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 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] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,5,2,4,3] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,3,2,4] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,5,4,2,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[3,1,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,1,2,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,1,5,2,4] => [2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,5,1,2,4] => [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[4,1,2,3,5] => [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[4,1,2,5,3] => [2,3,5,1,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[4,1,3,2,5] => [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[4,1,3,5,2] => [2,5,3,1,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[4,1,5,2,3] => [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[4,3,1,2,5] => [3,4,2,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[4,5,1,2,3] => [3,4,5,1,2] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,1,2,4,3] => [2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[5,1,3,2,4] => [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[5,1,3,4,2] => [2,5,3,4,1] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[5,1,4,2,3] => [2,4,5,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[5,1,4,3,2] => [2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,3,1,2,4] => [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[5,4,1,2,3] => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[5,4,1,3,2] => [3,5,4,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,4,3,1,2] => [4,5,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 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 oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
Map
to signed permutation
Description
The signed permutation with all signs positive.