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Identifier

St001865: Signed permutations ⟶ ℤ

Values

[1] => 1

[-1] => 0

[1,2] => 4

[1,-2] => 3

[-1,2] => 3

[-1,-2] => 2

[2,1] => 2

[2,-1] => 1

[-2,1] => 1

[-2,-1] => 0

[1,2,3] => 9

[1,2,-3] => 8

[1,-2,3] => 8

[1,-2,-3] => 7

[-1,2,3] => 8

[-1,2,-3] => 7

[-1,-2,3] => 7

[-1,-2,-3] => 6

[1,3,2] => 7

[1,3,-2] => 6

[1,-3,2] => 6

[1,-3,-2] => 5

[-1,3,2] => 6

[-1,3,-2] => 5

[-1,-3,2] => 5

[-1,-3,-2] => 4

[2,1,3] => 7

[2,1,-3] => 6

[2,-1,3] => 6

[2,-1,-3] => 5

[-2,1,3] => 6

[-2,1,-3] => 5

[-2,-1,3] => 5

[-2,-1,-3] => 4

[2,3,1] => 5

[2,3,-1] => 4

[2,-3,1] => 4

[2,-3,-1] => 3

[-2,3,1] => 4

[-2,3,-1] => 3

[-2,-3,1] => 3

[-2,-3,-1] => 2

[3,1,2] => 5

[3,1,-2] => 4

[3,-1,2] => 4

[3,-1,-2] => 3

[-3,1,2] => 4

[-3,1,-2] => 3

[-3,-1,2] => 3

[-3,-1,-2] => 2

[3,2,1] => 7

[3,2,-1] => 6

[3,-2,1] => 2

[3,-2,-1] => 1

[-3,2,1] => 6

[-3,2,-1] => 5

[-3,-2,1] => 1

[-3,-2,-1] => 0

[1,2,3,4] => 16

[1,2,3,-4] => 15

[1,2,-3,4] => 15

[1,2,-3,-4] => 14

[1,-2,3,4] => 15

[1,-2,3,-4] => 14

[1,-2,-3,4] => 14

[1,-2,-3,-4] => 13

[-1,2,3,4] => 15

[-1,2,3,-4] => 14

[-1,2,-3,4] => 14

[-1,2,-3,-4] => 13

[-1,-2,3,4] => 14

[-1,-2,3,-4] => 13

[-1,-2,-3,4] => 13

[-1,-2,-3,-4] => 12

[1,2,4,3] => 14

[1,2,4,-3] => 13

[1,2,-4,3] => 13

[1,2,-4,-3] => 12

[1,-2,4,3] => 13

[1,-2,4,-3] => 12

[1,-2,-4,3] => 12

[1,-2,-4,-3] => 11

[-1,2,4,3] => 13

[-1,2,4,-3] => 12

[-1,2,-4,3] => 12

[-1,2,-4,-3] => 11

[-1,-2,4,3] => 12

[-1,-2,4,-3] => 11

[-1,-2,-4,3] => 11

[-1,-2,-4,-3] => 10

[1,3,2,4] => 14

[1,3,2,-4] => 13

[1,3,-2,4] => 13

[1,3,-2,-4] => 12

[1,-3,2,4] => 13

[1,-3,2,-4] => 12

[1,-3,-2,4] => 12

[1,-3,-2,-4] => 11

[-1,3,2,4] => 13

[-1,3,2,-4] => 12

[-1,3,-2,4] => 12

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Description

The number of alignments of a signed permutation.
An alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $-n \leq i \leq j \leq n$, $i,j\neq 0$, such that one of the following conditions hold:

$i < j \leq \pi(j) < \pi(i)$, and $j > 0$ if it is a fixed point, or
$\pi(j) < \pi(i) \leq i < j)$ and $i < 0$ if it is a fixed point, or
$i \leq \pi(i) < \pi(j) \leq j$ and $i > 0$ if it is a fixed point and $j < 0$ if it is a fixed point, or
$\pi(i) \leq i < j \leq \pi(j)$ and $i < 0$ if it is a fixed point and $j > 0$ if it is a fixed point.

Let $al$ be the number of alignments of $\pi$, $cr$ be the number of crossings, St001862The number of crossings of a signed permutation., and let $fwex$ be the number of flag weak excedances, St001817The number of flag weak exceedances of a signed permutation.. Then
$$2 cr + al = n^2 - 2n + fwex.$$

References

[1] Corteel, S., Josuat-Vergès, M., Kim, J. S. Crossings of signed permutations and $q$-Eulerian numbers of type $B$ MathSciNet:3096133

Code

def statistic(pi):
    n = len(pi)
    al = []
    for j in range(-n, n+1):
        if j == 0:
            continue
        for i in range(-n, j):
            if i == 0:
                continue
            if ((i < j <= pi(j) < pi(i))
                and (j < pi(j) or j > 0)):
                al.append((i, j))
            elif ((pi(j) < pi(i) <= i < j)
                  and (pi(i) < i or i < 0)):
                al.append((i, j))
            elif ((i <= pi(i) < pi(j) <= j)
                  and (i < pi(i) or i > 0)
                  and (pi(j) < j or j < 0)):
                al.append((i, j))
            elif ((pi(i) <= i < j <= pi(j))
                  and (pi(i) < i or i < 0)
                  and (j < pi(j) or j > 0)):
                al.append((i, j))
    return len(al)

Created

Dec 01, 2022 at 11:07 by Martin Rubey

Updated

Dec 01, 2022 at 11:07 by Martin Rubey