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1. Definitions

2. Parking Functions and Dyck Paths

3. Examples

4. Properties

5. Conjectures

6. Statistics

We have the following 9 statistics on Parking functions in the database:

St000135 Parking functions ⟶ ℤ
The number of lucky cars of the parking function.
St000136 Parking functions ⟶ ℤ
The dinv of a parking function.
St000165 Parking functions ⟶ ℤ
Sum of the entries.
St000188 Parking functions ⟶ ℤ
The cosum of a parking function.
St000194 Parking functions ⟶ ℤ
The number of primary dinversion pairs of a labelled dyck path corresponding to a....
St000195 Parking functions ⟶ ℤ
The number of secondary dinversion pairs of the dyck path corresponding to a park....
St000540 Parking functions ⟶ ℤ
The sum of the entries of a parking function minus its length.
St000942 Parking functions ⟶ ℤ
The number of critical left to right maxima of the parking functions.
St000943 Parking functions ⟶ ℤ
The number of spots the most unlucky car had to go further in a parking function.....

7. Maps

We have the following 6 maps from and to Parking functions in the database:

Mp00052 Parking functions ⟶ Parking functions
to non-decreasing parking function
Mp00053 Parking functions ⟶ Permutations
to car permutation
Mp00054 Parking functions ⟶ Integer compositions
to inverse des composition
Mp00055 Parking functions ⟶ Permutations
to labelling permutation
Mp00056 Parking functions ⟶ Dyck paths
to Dyck path
Mp00057 Parking functions ⟶ Integer compositions
to touch composition

8. References

[CARLSSON]   E. Carlsson and A. Mellit, A Proof of the Shuffle Conjecture, http://arxiv.org/pdf/1508.06239v1.pdf.

[HICKS]   A. Hicks, A Parking Function Bijection Suggested by the Haglund-Morse-Zabrocki Conjecture, University of California- San Diego, (2010).

[STAN]   R. Stanley, Parking Functions, http://math.mit.edu/~rstan/transparencies/parking.pdf.