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Identifier

St001818: Decorated permutations ⟶ ℤ

Values

[+] => 0

[-] => 0

[+,+] => 0

[-,+] => 1

[+,-] => 1

[-,-] => 0

[2,1] => 0

[+,+,+] => 0

[-,+,+] => 2

[+,-,+] => 2

[+,+,-] => 2

[-,-,+] => 2

[-,+,-] => 2

[+,-,-] => 2

[-,-,-] => 0

[+,3,2] => 1

[-,3,2] => 1

[2,1,+] => 1

[2,1,-] => 1

[2,3,1] => 0

[3,1,2] => 0

[3,+,1] => 1

[3,-,1] => 1

[+,+,+,+] => 0

[-,+,+,+] => 3

[+,-,+,+] => 3

[+,+,-,+] => 3

[+,+,+,-] => 3

[-,-,+,+] => 4

[-,+,-,+] => 4

[-,+,+,-] => 4

[+,-,-,+] => 4

[+,-,+,-] => 4

[+,+,-,-] => 4

[-,-,-,+] => 3

[-,-,+,-] => 3

[-,+,-,-] => 3

[+,-,-,-] => 3

[-,-,-,-] => 0

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

[-,+,4,3] => 3

[+,-,4,3] => 3

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

[+,3,2,+] => 2

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

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

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

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

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

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

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

[+,4,+,2] => 2

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

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

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

[2,1,+,+] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[+,+,+,+,+] => 0

[-,+,+,+,+] => 4

[+,-,+,+,+] => 4

[+,+,-,+,+] => 4

[+,+,+,-,+] => 4

[+,+,+,+,-] => 4

[-,-,+,+,+] => 6

[-,+,-,+,+] => 6

[-,+,+,-,+] => 6

[-,+,+,+,-] => 6

[+,-,-,+,+] => 6

[+,-,+,-,+] => 6

[+,-,+,+,-] => 6

>>> Load all 1200 entries. <<<

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Description

The number of alignments of a decorated permutation.
An alignment in a decorated permutation $w$ is a pair of directed edges $(i\mapsto w(i), j\mapsto w(j))$ which can be drawn as distinct noncrossing arcs oriented in the same direction in the circular chord diagram.
According to [1], the number of alignments of a decorated permutation on $n$ letters with $k$ anit-exceedances equals and associated Grassmann interval $[u, v]$ equals $k(n - k) - [\ell(v) − \ell(u)]$.

References

[1] A. Postnikov, Total positivity, Grassmannians, and networks. 27 Sep 2006. Postnikov, A. Total positivity, Grassmannians, and networks arXiv:math/0609764
[2] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs arXiv:2207.06508

Code

# please simplify
def cyclic_interval(a, b, n):
    if a <= b:
        return list(range(a, b+1))
    return list(range(a, n+1)) + list(range(1, b+1))

def alignments(x):
    pi = x.to_signed_permutation().permutation()
    n = len(pi)
    al = []
    for i, e in enumerate(pi, 1):
        for j, f in enumerate(pi, 1):
            if i == j:
                continue
            a, b = sorted([i, e])
            c, d = sorted([j, f])
            if a <= c <= b <= d or c <= a <= d <= b:
                continue
            if (e not in cyclic_interval(i, f-1, n)
                or f not in cyclic_interval(e+1, j, n)):
                continue
            if j == f and x[j-1] < 0:
                continue
            if i == e and x[i-1] > 0:
                continue
            al.append((i, e, j, f))

    return al

def statistic(x):
    return len(alignments(x))

Created

Jul 15, 2022 at 18:19 by Martin Rubey

Updated

Jul 15, 2022 at 18:19 by Martin Rubey