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Identifier

St001822: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 0

[1,2] => 0

[1,-2] => 1

[-1,2] => 1

[-1,-2] => 1

[2,1] => 0

[2,-1] => 0

[-2,1] => 0

[-2,-1] => 0

[1,2,3] => 0

[1,2,-3] => 2

[1,-2,3] => 2

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

[-1,2,3] => 2

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

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

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

[1,3,2] => 1

[1,3,-2] => 1

[1,-3,2] => 2

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

[-1,3,2] => 2

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

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

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

[2,1,3] => 1

[2,1,-3] => 2

[2,-1,3] => 1

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

[-2,1,3] => 2

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

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

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

[2,3,1] => 0

[2,3,-1] => 0

[2,-3,1] => 1

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

[-2,3,1] => 1

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

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

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

[3,1,2] => 0

[3,1,-2] => 1

[3,-1,2] => 1

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

[-3,1,2] => 0

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

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

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

[3,2,1] => 1

[3,2,-1] => 1

[3,-2,1] => 0

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

[-3,2,1] => 2

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

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

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

[1,2,3,4] => 0

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

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

[1,2,-3,-4] => 5

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

[1,-2,3,-4] => 5

[1,-2,-3,4] => 5

[1,-2,-3,-4] => 6

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

[-1,2,3,-4] => 5

[-1,2,-3,4] => 5

[-1,2,-3,-4] => 6

[-1,-2,3,4] => 5

[-1,-2,3,-4] => 6

[-1,-2,-3,4] => 6

[-1,-2,-3,-4] => 6

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

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

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

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

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

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

[1,-2,-4,3] => 5

[1,-2,-4,-3] => 5

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

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

[-1,2,-4,3] => 5

[-1,2,-4,-3] => 5

[-1,-2,4,3] => 5

[-1,-2,4,-3] => 5

[-1,-2,-4,3] => 5

[-1,-2,-4,-3] => 5

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

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

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

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

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

[1,-3,2,-4] => 5

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

[1,-3,-2,-4] => 5

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

[-1,3,2,-4] => 5

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

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Description

The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, St001866The nesting alignments of a signed permutation., an alignment of type EN, St001867The number of alignments of type EN of a signed permutation., or an alignment of type NE, St001868The number of alignments of type NE of a signed permutation..
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, St001862The number of crossings of a signed permutation., let \operatorname{wex} be the number of weak excedances, St001863The number of weak excedances of a signed permutation., and let \operatorname{neg} be the number of negative entries, St001429The number of negative entries in a signed permutation.. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.

References

[1] Cho, S., Park, K. Permutation statistics and weak Bruhat order in permutation tableaux of type $B$ MathSciNet:3319075

Code

def statistic(pi):
    n = len(pi)
    A_nest = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
              if (-i < -j < -pi(j) < -pi(i))
              or (-i < j <= pi(j) < -pi(i))
              or (i < j <= pi(j) < pi(i))]
    A_EN = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
            if (-i < 0 < -pi(i) < pi(j) < j)
            or (i <= pi(i) < pi(j) < j)]
    A_NE = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
            if (pi(i) < i < j <= pi(j))]
    return len(A_nest + A_EN + A_NE)

Created

Jul 21, 2022 at 07:29 by Dennis Jahn

Updated

Dec 01, 2022 at 14:23 by Martin Rubey