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Identifier

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations

Images

[] => []

[1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 44392 entries. <<<

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

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map

$\Phi$ in [1, sec.3].

References

[1] Clarke, R. J., Steingr'ımsson, E., Zeng, J. New Euler-Mahonian statistics on permutations and words MathSciNet:1436481

Properties

bijective, graded

Sage code

def mapping(tau):
    tau = Permutation(tau)
    if not tau:
        return tau
    f = sorted([i+1 for i in range(len(tau)) if tau[i] > i+1])
    g = sorted([i+1 for i in range(len(tau)) if tau[i] <= i+1])

    F = set(f)
    G = set(g)
    F1 = set([tau[i-1] for i in F])
    G1 = set([tau[i-1] for i in G])

    openers = F.intersection(G1)
    closers = G.intersection(F1)
    insider = F.intersection(F1)
    outside = G.intersection(G1)

    assert len(list(openers) + list(closers) + list(insider) + list(outside)) == len(tau)

    # embracing numbers of pi are side numbers of tau
    E = []
    for i in range(len(tau)):
        if tau[i] > i+1:
            E.append(sum(1 for j in range(i) if tau[j] > tau[i]))
        else:
            E.append(sum(1 for j in range(i+1, len(tau)) if tau[j] < tau[i]))
    S = [] # Clarke's skeleton
    if 1 in outside:
        S.append([1])
    elif 1 in openers:
        S.append([1, infinity])
    tau_inv = tau.inverse()
    def find_open_block(e):
        i = len(S)-1
        while True:
            if infinity in S[i]:
                if e == 0:
                    assert i >= 0
                    return i
                e -= 1
            i -= 1
    for a in range(2, len(tau)+1):
        e = E[tau_inv[a-1]-1]
        if a in insider:
            i = find_open_block(e)
            S[i].append(a)
        elif a in closers:
            i = find_open_block(e)
            S[i].append(a)
            S[i].remove(infinity)
        elif a in outside:
            if e:
                i = find_open_block(e-1)
            else:
                i = len(S)
            S.insert(i, [a])
        elif a in openers:
            if e:
                i = find_open_block(e-1)
            else:
                i = len(S)            
            S.insert(i, [a, infinity])

    return Permutation(sum((sorted(s, reverse=True) for s in S), []))

Weight

Created

Sep 21, 2021 at 12:09 by Martin Rubey

Updated

Sep 21, 2021 at 12:09 by Martin Rubey