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Identifier

Mp00301: Parking functions —leading to repeats⟶ 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] => [1,2,2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> 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 prepending 0 and shifting the cumulative sums of a Prüfer code, then 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 $c_i = p_{i+1} - p_i \pmod{n+1}$.
The inverse is obtained by computing the cumulative sums, and then shifting all elements modulo $n+1$ such that the result is a parking function.
This is the inverse of Mp00298repeats to leading.
This map sends the number of occurrences of the first element to the number of repeated elements [3].

References

[1] Riordan, J. Ballots and trees MathSciNet:0234843
[2] Foata, D., Riordan, J. Mappings of acyclic and parking functions MathSciNet:0335294
[3] Stanley, R. P., Yin, M. Some enumerative properties of parking functions arXiv:2306.08681

Properties

bijective, graded

Sage code

def mapping(p):
    return pruefer_to_parking_pollak_riordan(parking_to_pruefer_circular(p))

def pruefer_to_parking_pollak_riordan(c):
    n = len(c) + 1
    pc = [sum(c[:i]) % (n+1) for i in range(n)]
    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_circular(p):
    n = len(p)
    d = p[0]
    return [(p[i] - d) % (n+1) for i in range(1, n)]

Weight

Created

Jun 16, 2023 at 23:23 by Martin Rubey

Updated

Jun 16, 2023 at 23:23 by Martin Rubey