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Identifier

Mp00298: Parking functions —repeats 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] => [2,3,1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> 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 computing successive differences to obtain a Prüfer code, then prepending 0 and cyclically shifting.
Pollak's map [1] sends a parking function

$(p_1,\dots,p_n)$ to the Prüfer code

$(c_1,\dots,c_{n-1})$ with

$c_i = p_{i+1} - p_i \pmod{n+1}$ .
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 repeated elements to the number of occurrences of the first element.

References

[1] Riordan, J. Ballots and trees MathSciNet:0234843
[2] Stanley, R. P., Yin, M. Some enumerative properties of parking functions arXiv:2306.08681

Properties

bijective, graded

Sage code

def mapping(p):
    if not p:
        return p
    return pruefer_to_parking_circular(parking_to_pruefer_pollak_riordan(p))

def parking_to_pruefer_pollak_riordan(p):
    n = len(p)
    return [(p[i+1] - p[i]) % (n+1) for i in range(n-1)]

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)

Weight

Created

Jun 16, 2023 at 13:04 by Martin Rubey

Updated

Jun 16, 2023 at 22:44 by Martin Rubey