Processing math: 100%

Identifier
Values
[1,2] => [1,2] => [2,-1] => [1,2] => 0
[2,1] => [2,1] => [1,-2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,-1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,-1] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1,2,3] => 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,1,-2] => [2,1,3] => 0
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1,3,2,4] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [2,1,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [3,1,2,4] => 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [2,3,1,4] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [3,2,1,4] => 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,2,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,3,2,4] => 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,3,4] => 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,4] => 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,2,4] => 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,1,3,4] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,3,4] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,4] => 0
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [3,1,2,4] => 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,4] => 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,2,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [2,3,1,4] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [2,3,1,4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,1,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1,2,4,3,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,4,2,3,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [1,3,4,2,5] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [1,4,3,2,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [2,1,3,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [3,1,2,4,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [4,1,2,3,5] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [3,1,4,2,5] => 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [4,1,3,2,5] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [2,3,1,4,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [2,4,1,3,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [3,2,1,4,5] => 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [4,2,1,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [3,4,1,2,5] => 2
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [4,3,1,2,5] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2,3,4,1,5] => 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => [2,4,3,1,5] => 1
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => [3,2,4,1,5] => 1
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [4,2,3,1,5] => 2
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [3,4,2,1,5] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [4,3,2,1,5] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,2,4,3,5] => 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,3,2,4,5] => 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,4,2,3,5] => 1
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1,3,4,2,5] => 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [1,4,3,2,5] => 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,2,4,3,5] => 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,2,4,3,5] => 0
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [1,2,4,3,5] => 0
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1,3,2,4,5] => 0
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,4,2,3,5] => 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [1,3,2,4,5] => 0
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [1,3,2,4,5] => 0
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1,4,2,3,5] => 1
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [1,4,2,3,5] => 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [1,3,4,2,5] => 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1,4,3,2,5] => 1
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1,3,4,2,5] => 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [1,3,4,2,5] => 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,4,3,2,5] => 1
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [1,4,3,2,5] => 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => [2,1,3,4,5] => 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [2,1,4,3,5] => 0
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [3,1,2,4,5] => 1
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [4,1,2,3,5] => 1
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => [3,1,4,2,5] => 0
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [4,1,3,2,5] => 1
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2,1,3,4,5] => 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1,4,3,5] => 0
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1,3,4,5] => 0
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,3,4,5] => 0
[3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,-4] => [2,1,4,3,5] => 0
[3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => [2,1,4,3,5] => 0
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => [3,1,2,4,5] => 1
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [4,1,2,3,5] => 1
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => [3,1,2,4,5] => 1
[3,4,2,5,1] => [3,4,2,5,1] => [3,1,2,4,-5] => [3,1,2,4,5] => 1
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => [4,1,2,3,5] => 1
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => [4,1,2,3,5] => 1
[3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => [3,1,4,2,5] => 0
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => [4,1,3,2,5] => 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => [3,1,4,2,5] => 0
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => [3,1,4,2,5] => 0
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [4,1,3,2,5] => 1
[3,5,4,2,1] => [3,5,4,2,1] => [4,1,3,2,-5] => [4,1,3,2,5] => 1
[4,1,2,3,5] => [4,1,2,3,5] => [3,4,1,5,-2] => [2,3,1,4,5] => 1
[4,1,2,5,3] => [4,1,2,5,3] => [3,5,1,4,-2] => [2,4,1,3,5] => 2
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => [3,2,1,4,5] => 1
[4,1,3,5,2] => [4,1,3,5,2] => [5,3,1,4,-2] => [4,2,1,3,5] => 1
[4,1,5,2,3] => [4,1,5,2,3] => [4,5,1,3,-2] => [3,4,1,2,5] => 2
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => [4,3,1,2,5] => 2
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => [2,3,1,4,5] => 1
[4,2,1,5,3] => [4,2,1,5,3] => [2,5,1,4,-3] => [2,4,1,3,5] => 2
[4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,5,-4] => [2,3,1,4,5] => 1
[4,2,3,5,1] => [4,2,3,5,1] => [2,3,1,4,-5] => [2,3,1,4,5] => 1
[4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => [2,4,1,3,5] => 2
[4,2,5,3,1] => [4,2,5,3,1] => [2,4,1,3,-5] => [2,4,1,3,5] => 2
[4,3,1,2,5] => [4,3,1,2,5] => [4,2,1,5,-3] => [3,2,1,4,5] => 1
[4,3,1,5,2] => [4,3,1,5,2] => [5,2,1,4,-3] => [4,2,1,3,5] => 1
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,-4] => [3,2,1,4,5] => 1
[4,3,2,5,1] => [4,3,2,5,1] => [3,2,1,4,-5] => [3,2,1,4,5] => 1
[4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => [4,2,1,3,5] => 1
[4,3,5,2,1] => [4,3,5,2,1] => [4,2,1,3,-5] => [4,2,1,3,5] => 1
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => [3,4,1,2,5] => 2
[4,5,1,3,2] => [4,5,1,3,2] => [5,4,1,2,-3] => [4,3,1,2,5] => 2
[4,5,2,1,3] => [4,5,2,1,3] => [3,5,1,2,-4] => [3,4,1,2,5] => 2
[4,5,2,3,1] => [4,5,2,3,1] => [3,4,1,2,-5] => [3,4,1,2,5] => 2
[4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => [4,3,1,2,5] => 2
[4,5,3,2,1] => [4,5,3,2,1] => [4,3,1,2,-5] => [4,3,1,2,5] => 2
[5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => [2,3,4,1,5] => 1
[5,1,2,4,3] => [5,1,2,4,3] => [3,5,4,1,-2] => [2,4,3,1,5] => 1
[5,1,3,2,4] => [5,1,3,2,4] => [4,3,5,1,-2] => [3,2,4,1,5] => 1
[5,1,3,4,2] => [5,1,3,4,2] => [5,3,4,1,-2] => [4,2,3,1,5] => 2
[5,1,4,2,3] => [5,1,4,2,3] => [4,5,3,1,-2] => [3,4,2,1,5] => 2
[5,1,4,3,2] => [5,1,4,3,2] => [5,4,3,1,-2] => [4,3,2,1,5] => 2
[5,2,1,3,4] => [5,2,1,3,4] => [2,4,5,1,-3] => [2,3,4,1,5] => 1
[5,2,1,4,3] => [5,2,1,4,3] => [2,5,4,1,-3] => [2,4,3,1,5] => 1
[5,2,3,1,4] => [5,2,3,1,4] => [2,3,5,1,-4] => [2,3,4,1,5] => 1
[5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,1,-5] => [2,3,4,1,5] => 1
[5,2,4,1,3] => [5,2,4,1,3] => [2,5,3,1,-4] => [2,4,3,1,5] => 1
[5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,1,-5] => [2,4,3,1,5] => 1
[5,3,1,2,4] => [5,3,1,2,4] => [4,2,5,1,-3] => [3,2,4,1,5] => 1
[5,3,1,4,2] => [5,3,1,4,2] => [5,2,4,1,-3] => [4,2,3,1,5] => 2
[5,3,2,1,4] => [5,3,2,1,4] => [3,2,5,1,-4] => [3,2,4,1,5] => 1
[5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,1,-5] => [3,2,4,1,5] => 1
[5,3,4,1,2] => [5,3,4,1,2] => [5,2,3,1,-4] => [4,2,3,1,5] => 2
[5,3,4,2,1] => [5,3,4,2,1] => [4,2,3,1,-5] => [4,2,3,1,5] => 2
[5,4,1,2,3] => [5,4,1,2,3] => [4,5,2,1,-3] => [3,4,2,1,5] => 2
[5,4,1,3,2] => [5,4,1,3,2] => [5,4,2,1,-3] => [4,3,2,1,5] => 2
[5,4,2,1,3] => [5,4,2,1,3] => [3,5,2,1,-4] => [3,4,2,1,5] => 2
[5,4,2,3,1] => [5,4,2,3,1] => [3,4,2,1,-5] => [3,4,2,1,5] => 2
[5,4,3,1,2] => [5,4,3,1,2] => [5,3,2,1,-4] => [4,3,2,1,5] => 2
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,-5] => [4,3,2,1,5] => 2
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
click to show known generating functions       
Description
The number of descents of distance 2 of a permutation.
This is, des2(π)=|{i:π(i)>π(i+2)}|.
Map
inverse Kreweras complement
Description
The inverse Kreweras complement of a signed permutation.
This is the signed permutation cπ−1 where c=(1,…,n,−1,−2,…,−n) is the long cycle.
The order of the inverse Kreweras complement on signed permutations of {±1,…,±n} is 2n.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
standardize
Description
Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let π∈Hn be a signed permutation. Assuming the order 1<⋯<n<−n<⋯<−1, this map returns the permutation in Sn which is order isomorphic to π(1),…,π(n).