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Identifier

Mp00271: Signed permutations —flag negatives to negatives⟶ Signed permutations

Images

[1] => [1]

[-1] => [-1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

A map sending the number of flag negative entries to the number of negative entries.
For a signed permutation $\pi\in\mathfrak H_n$, a flag negative entry is a positive entry followed by a negative entry in the sequence $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$, see St001870The number of positive entries followed by a negative entry in a signed permutation..
This map leaves the underlying permutation intact, and uses Mp00269flag zeros to zeros on the sequence of signs.

References

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

Properties

bijective, graded

Sage code

def mp00269(w):
    zeros = []
    letter = 1
    z = 0
    for e in reversed(w):
        if e == letter:
            z += 1
        else:
            zeros.append(z)
            z = 0
            letter = 1 - letter
    result = [1]*len(w)
    last = -1
    for z in zeros:
        last = z + last + 1
        result[last] = 0
    return Words([0,1])(result)

def mapping(pi):
    w = pi.permutation()
    s = mp00269([1 if p > 0 else 0 for p in pi])
    return SignedPermutation([p if e else -p for e, p in zip(s, w)])

Weight

Created

Dec 04, 2022 at 20:22 by Martin Rubey

Updated

Dec 04, 2022 at 20:22 by Martin Rubey