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Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations

Images

[] => []

[1,0] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1]

[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5]

[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4]

[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4]

[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5]

[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3]

[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5]

[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3]

[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3]

[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5]

[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4]

[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4]

[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4]

[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5]

[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2]

[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5]

[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4]

[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4]

[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5]

[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2]

[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5]

[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2]

[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2]

[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5]

[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4]

[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4]

[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4]

[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5]

[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3]

[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5]

[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3]

[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3]

[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5]

[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3]

[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3]

[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3]

[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5]

[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4]

[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4]

[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4]

[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4]

[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5]

[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1]

[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6]

[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5]

[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5]

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6]

[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4]

[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6]

[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4]

[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4]

[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6]

[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5]

[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5]

[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5]

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6]

[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3]

[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6]

[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5]

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5]

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6]

[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3]

[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6]

[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3]

[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3]

[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6]

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5]

[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5]

[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5]

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6]

[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4]

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6]

[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4]

[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4]

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6]

[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4]

[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4]

[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4]

>>> Load all 6094 entries. <<<

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

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

References

[1] Billey, S. C., Jockusch, W., Stanley, R. P. Some combinatorial properties of Schubert polynomials MathSciNet:1241505
[2] Callan, D. Bijections from Dyck paths to 321-avoiding permutations revisited arXiv:0711.2684

Properties

injective, graded

Sage code

def mapping(P):
    """
    sage: sorted([mapping(D) for D in DyckWords(7)]) == sorted(Permutations(7, avoiding=[3,2,1]))
    True
    """
    n = (len(P)//2)
    a = P.rise_composition()
    A = [1+sum(a[:i]) for i in range(1,len(a))]
    d = P.reverse().rise_composition().reversed()
    D = [sum(d[:i])-1 for i in range(1,len(a))]
    pi = [None]*n
    rest = list(range(1, n+1))
    for (e,f) in zip(A, D):
        pi[f] = e
        rest.remove(e)

    i = 0
    for e in rest:
        while pi[i] is not None:
            i += 1
        pi[i] = e
        
    return Permutation(pi)

Weight

Created

Jan 29, 2020 at 13:27 by FindStatCrew

Updated

Jan 29, 2020 at 13:27 by Martin Rubey