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Identifier

Mp00300: Parking functions —leading to ones⟶ Parking functions

Images

[1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,4] => [1,1,1,4]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 18248 entries. <<<

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

Description

The unique parking function obtained by subtracting the first entry from all others to obtain a Prüfer code and then applying the inverse of the Foata-Riordan map.
This map is the inverse of Mp00299ones to leading.

References

[1] Foata, D., Riordan, J. Mappings of acyclic and parking functions MathSciNet:0335294

Properties

bijective, graded

Sage code

def mapping(p):
    if not p:
        return p
    return pruefer_to_parking_foata_riordan(parking_to_pruefer_circular(p))

def parking_to_pruefer_circular(p):
    n = len(p)
    d = p[0]
    return [(p[i] - d) % (n+1) for i in range(1, n)]

def pruefer_to_tree(c):
    c = list(c)
    n = len(c) 
    V = list(range(n+2))
    G = Graph([V, []])
    for _ in range(n):
        i, u = next((i, u) for i, u in enumerate(V) if u not in c)
        del V[i]
        v = c.pop(0)
        G.add_edge(u, v)
    G.add_edge(V)
    return to_tree(G, 0)

def to_tree(G, r):
    if not G:
        return LabelledRootedTree([], r)
    roots = G[r]
    G.delete_vertex(r)
    return LabelledOrderedTree([to_tree(G.subgraph(G.connected_component_containing_vertex(u)), u)
                               for u in roots], r)


def pruefer_to_parking_foata_riordan(c):
    t = pruefer_to_tree(c)
    n = len(c) + 2
    labels = []
    f = [None]*(n-1)
    def action(u):
        i = u.label()
        labels.append(i)
        for c in u:
            f[c.label()-1] = i
    t.breadth_first_order_traversal(action)
    sigma = Permutation(labels[1:]).inverse()
    g = []
    for e in f:
        if e:
            g.append(1 + sigma(e))
        else:
            g.append(1)
    return ParkingFunction(g)

Weight

Created

Jun 16, 2023 at 22:35 by Martin Rubey

Updated

Jun 16, 2023 at 22:35 by Martin Rubey