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Identifier

Mp00194: Signed permutations —Foata-Han inverse⟶ 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] => [-3,1,2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 13423 entries. <<<

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

Description

The inverse of the Foata-Han bijection for signed permutations.
This map sends the the flag inversion number St001428The number of B-inversions of a signed permutation. to the flag major index St001433The flag major index of a signed permutation.

References

[1] Foata, D., Han, G.-N. Signed words and permutations. I. A fundamental transformation MathSciNet:2280171

Properties

bijective, graded

Sage code

def right_to_left_factorization(w, x):
    result = []
    v = []
    if w[-1] <= x:
        for y in w:
            if y <= x:
                result.append((v, y))
                v = []
            else:
                v.append(y)
    else:
        for y in w:
            if y > x:
                result.append((v, y))
                v = []
            else:
                v.append(y)
    return result

def gamma(w, x):
    """
    sage: gamma([3], -2)
    [3]

    sage: gamma([-3, -2], 1)
    [-3, -2]

    sage: gamma([-2, 3, 1], -3)
    [-2, 3, 1]

    sage: gamma([3, -2, 1, -3, -4], 3)
    [3, -2, 1, -3, -4]
    """
    vy = right_to_left_factorization(w, x)
    return sum([[y] + v for v, y in vy], [])

def gamma_inverse(w, x):
    """
    sage: n=5; 
    sage: all(all(gamma_inverse(gamma(list(pi), x), x) == list(pi) for pi in SignedPermutations(n)) for x in range(-n,n+1))
    True
    """
    return delta(w, x+1)

def left_to_right_factorization(w, x):
    """
    sage: left_to_right_factorization([3, -2], -1)
    [(3, [-2])]
    """
    result = []
    v = []
    y0 = None
    if w[0] >= x:
        for y in w:
            if y >= x:
                if y0 is not None:
                    result.append((y0, v))
                y0 = y
                v = []
            else:
                v.append(y)
    else:
        for y in w:
            if y < x:
                if y0 is not None:
                    result.append((y0, v))                
                y0 = y
                v = []
            else:
                v.append(y)
    if y0 is not None:
        result.append((y0, v))
    return result

def delta(w, x):
    """
    sage: delta([-3], 2)
    [-3]

    sage: delta([3, -2], -1)
    [-2, 3]

    sage: delta([-2, 3, 1], 3)
    [3, -2, 1]

    sage: delta([3, -2, 1, -3, -4], -3)
    [3, -2, 1, -4, -3]
    """
    yv = left_to_right_factorization(w, x)
    return sum([v + [y] for y, v in yv], [])

def delta_inverse(w, x):
    """
    sage: n=5; 
    sage: all(all(delta_inverse(delta(list(pi), x), x) == list(pi) for pi in SignedPermutations(n)) for x in range(-n,n+1))
    True
    """
    return gamma(w, x-1)

def s1(w):
    return [-w[0]] + w[1:]

def Psi_inverse(w):
    if len(w) <= 1:
        return w
    x = w.pop()
    v_prime = delta_inverse(w, -x)
    z1 = v_prime[0]
    if -x <= z1 <= x or x < z1 < -x:
        return Psi_inverse(gamma_inverse(delta_inverse(w, -x), x)) + [x]
    return Psi_inverse(gamma_inverse(s1(delta_inverse(w, -x)), x)) + [x]

def mapping(pi):
    return pi.parent()(Psi_inverse(list(pi)))

Weight

Created

Sep 27, 2020 at 21:49 by Martin Rubey

Updated

Sep 27, 2020 at 21:49 by Martin Rubey