Identifier
Identifier
Values
[+] => 0
[-] => 1
[+,+] => 0
[-,+] => 1
[+,-] => 1
[-,-] => 2
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 1
[+,-,+] => 1
[+,+,-] => 1
[-,-,+] => 2
[-,+,-] => 2
[+,-,-] => 2
[-,-,-] => 3
[+,3,2] => 0
[-,3,2] => 1
[2,1,+] => 0
[2,1,-] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 0
[3,-,1] => 1
[+,+,+,+] => 0
[-,+,+,+] => 1
[+,-,+,+] => 1
[+,+,-,+] => 1
[+,+,+,-] => 1
[-,-,+,+] => 2
[-,+,-,+] => 2
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 2
[+,+,-,-] => 2
[-,-,-,+] => 3
[-,-,+,-] => 3
[-,+,-,-] => 3
[+,-,-,-] => 3
[-,-,-,-] => 4
[+,+,4,3] => 0
[-,+,4,3] => 1
[+,-,4,3] => 1
[-,-,4,3] => 2
[+,3,2,+] => 0
[-,3,2,+] => 1
[+,3,2,-] => 1
[-,3,2,-] => 2
[+,3,4,2] => 0
[-,3,4,2] => 1
[+,4,2,3] => 0
[-,4,2,3] => 1
[+,4,+,2] => 0
[-,4,+,2] => 1
[+,4,-,2] => 1
[-,4,-,2] => 2
[2,1,+,+] => 0
[2,1,-,+] => 1
[2,1,+,-] => 1
[2,1,-,-] => 2
[2,1,4,3] => 0
[2,3,1,+] => 0
[2,3,1,-] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,+,1] => 0
[2,4,-,1] => 1
[3,1,2,+] => 0
[3,1,2,-] => 1
[3,1,4,2] => 0
[3,+,1,+] => 0
[3,-,1,+] => 1
[3,+,1,-] => 1
[3,-,1,-] => 2
[3,+,4,1] => 0
[3,-,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,+,2] => 0
[4,1,-,2] => 1
[4,+,1,3] => 0
[4,-,1,3] => 1
[4,+,+,1] => 0
[4,-,+,1] => 1
[4,+,-,1] => 1
[4,-,-,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 0
[+,+,+,+,+] => 0
[-,+,+,+,+] => 1
[+,-,+,+,+] => 1
[+,+,-,+,+] => 1
[+,+,+,-,+] => 1
[+,+,+,+,-] => 1
[-,-,+,+,+] => 2
[-,+,-,+,+] => 2
[-,+,+,-,+] => 2
[-,+,+,+,-] => 2
[+,-,-,+,+] => 2
[+,-,+,-,+] => 2
[+,-,+,+,-] => 2
[+,+,-,-,+] => 2
[+,+,-,+,-] => 2
[+,+,+,-,-] => 2
[-,-,-,+,+] => 3
[-,-,+,-,+] => 3
[-,-,+,+,-] => 3
[-,+,-,-,+] => 3
[-,+,-,+,-] => 3
[-,+,+,-,-] => 3
[+,-,-,-,+] => 3
[+,-,-,+,-] => 3
[+,-,+,-,-] => 3
[+,+,-,-,-] => 3
[-,-,-,-,+] => 4
[-,-,-,+,-] => 4
[-,-,+,-,-] => 4
[-,+,-,-,-] => 4
[+,-,-,-,-] => 4
[-,-,-,-,-] => 5
[+,+,+,5,4] => 0
[-,+,+,5,4] => 1
[+,-,+,5,4] => 1
[+,+,-,5,4] => 1
[-,-,+,5,4] => 2
[-,+,-,5,4] => 2
[+,-,-,5,4] => 2
[-,-,-,5,4] => 3
[+,+,4,3,+] => 0
[-,+,4,3,+] => 1
[+,-,4,3,+] => 1
[+,+,4,3,-] => 1
[-,-,4,3,+] => 2
[-,+,4,3,-] => 2
[+,-,4,3,-] => 2
[-,-,4,3,-] => 3
[+,+,4,5,3] => 0
[-,+,4,5,3] => 1
[+,-,4,5,3] => 1
[-,-,4,5,3] => 2
[+,+,5,3,4] => 0
[-,+,5,3,4] => 1
[+,-,5,3,4] => 1
[-,-,5,3,4] => 2
[+,+,5,+,3] => 0
[-,+,5,+,3] => 1
[+,-,5,+,3] => 1
[+,+,5,-,3] => 1
[-,-,5,+,3] => 2
[-,+,5,-,3] => 2
[+,-,5,-,3] => 2
[-,-,5,-,3] => 3
[+,3,2,+,+] => 0
[-,3,2,+,+] => 1
[+,3,2,-,+] => 1
[+,3,2,+,-] => 1
[-,3,2,-,+] => 2
[-,3,2,+,-] => 2
[+,3,2,-,-] => 2
[-,3,2,-,-] => 3
[+,3,2,5,4] => 0
[-,3,2,5,4] => 1
[+,3,4,2,+] => 0
[-,3,4,2,+] => 1
[+,3,4,2,-] => 1
[-,3,4,2,-] => 2
[+,3,4,5,2] => 0
[-,3,4,5,2] => 1
[+,3,5,2,4] => 0
[-,3,5,2,4] => 1
[+,3,5,+,2] => 0
[-,3,5,+,2] => 1
[+,3,5,-,2] => 1
[-,3,5,-,2] => 2
[+,4,2,3,+] => 0
[-,4,2,3,+] => 1
[+,4,2,3,-] => 1
[-,4,2,3,-] => 2
[+,4,2,5,3] => 0
[-,4,2,5,3] => 1
[+,4,+,2,+] => 0
[-,4,+,2,+] => 1
[+,4,-,2,+] => 1
[+,4,+,2,-] => 1
[-,4,-,2,+] => 2
[-,4,+,2,-] => 2
[+,4,-,2,-] => 2
[-,4,-,2,-] => 3
[+,4,+,5,2] => 0
[-,4,+,5,2] => 1
[+,4,-,5,2] => 1
[-,4,-,5,2] => 2
[+,4,5,2,3] => 0
[-,4,5,2,3] => 1
[+,4,5,3,2] => 0
[-,4,5,3,2] => 1
[+,5,2,3,4] => 0
[-,5,2,3,4] => 1
[+,5,2,+,3] => 0
[-,5,2,+,3] => 1
[+,5,2,-,3] => 1
[-,5,2,-,3] => 2
[+,5,+,2,4] => 0
[-,5,+,2,4] => 1
[+,5,-,2,4] => 1
[-,5,-,2,4] => 2
[+,5,+,+,2] => 0
[-,5,+,+,2] => 1
[+,5,-,+,2] => 1
[+,5,+,-,2] => 1
[-,5,-,+,2] => 2
[-,5,+,-,2] => 2
[+,5,-,-,2] => 2
[-,5,-,-,2] => 3
[+,5,4,2,3] => 0
[-,5,4,2,3] => 1
[+,5,4,3,2] => 0
[-,5,4,3,2] => 1
[2,1,+,+,+] => 0
[2,1,-,+,+] => 1
[2,1,+,-,+] => 1
[2,1,+,+,-] => 1
[2,1,-,-,+] => 2
[2,1,-,+,-] => 2
[2,1,+,-,-] => 2
[2,1,-,-,-] => 3
[2,1,+,5,4] => 0
[2,1,-,5,4] => 1
[2,1,4,3,+] => 0
[2,1,4,3,-] => 1
[2,1,4,5,3] => 0
[2,1,5,3,4] => 0
[2,1,5,+,3] => 0
[2,1,5,-,3] => 1
[2,3,1,+,+] => 0
[2,3,1,-,+] => 1
[2,3,1,+,-] => 1
[2,3,1,-,-] => 2
[2,3,1,5,4] => 0
[2,3,4,1,+] => 0
[2,3,4,1,-] => 1
[2,3,4,5,1] => 0
[2,3,5,1,4] => 0
[2,3,5,+,1] => 0
[2,3,5,-,1] => 1
[2,4,1,3,+] => 0
[2,4,1,3,-] => 1
[2,4,1,5,3] => 0
[2,4,+,1,+] => 0
[2,4,-,1,+] => 1
[2,4,+,1,-] => 1
[2,4,-,1,-] => 2
[2,4,+,5,1] => 0
[2,4,-,5,1] => 1
[2,4,5,1,3] => 0
[2,4,5,3,1] => 0
[2,5,1,3,4] => 0
[2,5,1,+,3] => 0
[2,5,1,-,3] => 1
[2,5,+,1,4] => 0
[2,5,-,1,4] => 1
[2,5,+,+,1] => 0
[2,5,-,+,1] => 1
[2,5,+,-,1] => 1
[2,5,-,-,1] => 2
[2,5,4,1,3] => 0
[2,5,4,3,1] => 0
[3,1,2,+,+] => 0
[3,1,2,-,+] => 1
[3,1,2,+,-] => 1
[3,1,2,-,-] => 2
[3,1,2,5,4] => 0
[3,1,4,2,+] => 0
[3,1,4,2,-] => 1
[3,1,4,5,2] => 0
[3,1,5,2,4] => 0
[3,1,5,+,2] => 0
[3,1,5,-,2] => 1
[3,+,1,+,+] => 0
[3,-,1,+,+] => 1
[3,+,1,-,+] => 1
[3,+,1,+,-] => 1
[3,-,1,-,+] => 2
[3,-,1,+,-] => 2
[3,+,1,-,-] => 2
[3,-,1,-,-] => 3
[3,+,1,5,4] => 0
[3,-,1,5,4] => 1
[3,+,4,1,+] => 0
[3,-,4,1,+] => 1
[3,+,4,1,-] => 1
[3,-,4,1,-] => 2
[3,+,4,5,1] => 0
[3,-,4,5,1] => 1
[3,+,5,1,4] => 0
[3,-,5,1,4] => 1
[3,+,5,+,1] => 0
[3,-,5,+,1] => 1
[3,+,5,-,1] => 1
[3,-,5,-,1] => 2
[3,4,1,2,+] => 0
[3,4,1,2,-] => 1
[3,4,1,5,2] => 0
[3,4,2,1,+] => 0
[3,4,2,1,-] => 1
[3,4,2,5,1] => 0
[3,4,5,1,2] => 0
[3,4,5,2,1] => 0
[3,5,1,2,4] => 0
[3,5,1,+,2] => 0
[3,5,1,-,2] => 1
[3,5,2,1,4] => 0
[3,5,2,+,1] => 0
[3,5,2,-,1] => 1
[3,5,4,1,2] => 0
[3,5,4,2,1] => 0
[4,1,2,3,+] => 0
[4,1,2,3,-] => 1
[4,1,2,5,3] => 0
[4,1,+,2,+] => 0
[4,1,-,2,+] => 1
[4,1,+,2,-] => 1
[4,1,-,2,-] => 2
[4,1,+,5,2] => 0
[4,1,-,5,2] => 1
[4,1,5,2,3] => 0
[4,1,5,3,2] => 0
[4,+,1,3,+] => 0
[4,-,1,3,+] => 1
[4,+,1,3,-] => 1
[4,-,1,3,-] => 2
[4,+,1,5,3] => 0
[4,-,1,5,3] => 1
[4,+,+,1,+] => 0
[4,-,+,1,+] => 1
[4,+,-,1,+] => 1
[4,+,+,1,-] => 1
[4,-,-,1,+] => 2
[4,-,+,1,-] => 2
[4,+,-,1,-] => 2
[4,-,-,1,-] => 3
[4,+,+,5,1] => 0
[4,-,+,5,1] => 1
[4,+,-,5,1] => 1
[4,-,-,5,1] => 2
[4,+,5,1,3] => 0
[4,-,5,1,3] => 1
[4,+,5,3,1] => 0
[4,-,5,3,1] => 1
[4,3,1,2,+] => 0
[4,3,1,2,-] => 1
[4,3,1,5,2] => 0
[4,3,2,1,+] => 0
[4,3,2,1,-] => 1
[4,3,2,5,1] => 0
[4,3,5,1,2] => 0
[4,3,5,2,1] => 0
[4,5,1,2,3] => 0
[4,5,1,3,2] => 0
[4,5,2,1,3] => 0
[4,5,2,3,1] => 0
[4,5,+,1,2] => 0
[4,5,-,1,2] => 1
[4,5,+,2,1] => 0
[4,5,-,2,1] => 1
[5,1,2,3,4] => 0
[5,1,2,+,3] => 0
[5,1,2,-,3] => 1
[5,1,+,2,4] => 0
[5,1,-,2,4] => 1
[5,1,+,+,2] => 0
[5,1,-,+,2] => 1
[5,1,+,-,2] => 1
[5,1,-,-,2] => 2
[5,1,4,2,3] => 0
[5,1,4,3,2] => 0
[5,+,1,3,4] => 0
[5,-,1,3,4] => 1
[5,+,1,+,3] => 0
[5,-,1,+,3] => 1
[5,+,1,-,3] => 1
[5,-,1,-,3] => 2
[5,+,+,1,4] => 0
[5,-,+,1,4] => 1
[5,+,-,1,4] => 1
[5,-,-,1,4] => 2
[5,+,+,+,1] => 0
[5,-,+,+,1] => 1
[5,+,-,+,1] => 1
[5,+,+,-,1] => 1
[5,-,-,+,1] => 2
[5,-,+,-,1] => 2
[5,+,-,-,1] => 2
[5,-,-,-,1] => 3
[5,+,4,1,3] => 0
[5,-,4,1,3] => 1
[5,+,4,3,1] => 0
[5,-,4,3,1] => 1
[5,3,1,2,4] => 0
[5,3,1,+,2] => 0
[5,3,1,-,2] => 1
[5,3,2,1,4] => 0
[5,3,2,+,1] => 0
[5,3,2,-,1] => 1
[5,3,4,1,2] => 0
[5,3,4,2,1] => 0
[5,4,1,2,3] => 0
[5,4,1,3,2] => 0
[5,4,2,1,3] => 0
[5,4,2,3,1] => 0
[5,4,+,1,2] => 0
[5,4,-,1,2] => 1
[5,4,+,2,1] => 0
[5,4,-,2,1] => 1
click to show generating function       
Description
The number of negatively decorated fixed points of a decorated permutation.
Code
def as_permutation(pi):
    pi = list(pi)
    for i,a in enumerate(pi):
        if a < 0:
            pi[i] = -a
    return Permutation(pi)

def statistic(pi):
    tau = list(pi)
    return sum(1 for i in as_permutation(pi).fixed_points() if tau[i-1] < 0)
Created
Jun 20, 2019 at 08:00 by Christian Stump
Updated
Jun 20, 2019 at 08:00 by Christian Stump