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Identifier

Mp00299: Parking functions —ones to leading⟶ 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] => [1,3,1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 applying the Foata-Riordan map to obtain a Prüfer code, then prepending 0 and cyclically shifting.
Let

$c_1,\dots,c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let

$c_0=0$ . This map returns the unique parking function

$(q_1,\dots,q_n)$ such that

$q_i-c_{i-1}$ is constant modulo

$n+1$ , see [2, sec 4].
This map sends the number of ones to the number of occurrences of the first element.

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_circular(parking_to_pruefer_foata_riordan(p))

def pruefer_to_parking_circular(c):
    pc = [0] + list(c)
    n = len(pc)
    P = ParkingFunctions(n)
    for i in range(n):
        if pc in P:
            break
        pc = [(e + 1) % (n+1) for e in pc]
    return ParkingFunction(pc)

def parking_to_pruefer_foata_riordan(p):
    n = len(p)
    tau = Permutation([sum(1 for j in range(n)
                           if p[j] < p[i] or (p[j] == p[i] and j <= i))
                      for i in range(n)])
    pi = tau.inverse()
    E = [(0, i) if c == 1 else (i, pi(c-1)) for i, c in enumerate(p, 1)]
    V = list(range(n+1))
    G = Graph([V, E])
    c = []
    for _ in range(n-1):
        for u in V:
            if G.degree(u) == 1:
                c.append(G[u][0])
                G.delete_vertex(u)
                V.remove(u)
                break
    return c

Weight

Created

Jun 16, 2023 at 15:26 by Martin Rubey

Updated

Feb 01, 2025 at 16:08 by Martin Rubey