Parking Functions¶

INFORMALLY (reference [Beck]):

Imagine a one-way cul-de-sac with $$n$$ parking spots. We will give the first parking spot the number 1, the next one number 2, etc., down to the last one, number $$n$$. Initially they are all free, but there are $$n$$ cars approaching the street, and they would all like to park there. To make life interesting, every car has a parking preference, and we record the preferences in a sequence; For example, if $$n = 3$$, the sequence $$(2, 1, 1)$$ means that the first car would like to park at spot number 2, the second car prefers parking spot number 1, and the last car would also like to part at number 1. The street is very narrow, so there is no way to back up. Now each car enters the street and approaches its preferred parking spot; if it is free, it parks there, and if not, it moves down the street to the first available spot. We call a sequence a parking function (of length $$n$$) if all cars end up finding a parking spot. For example, the sequence $$(2, 1, 1)$$ is a parking sequence (of length 3), whereas the sequence $$(2, 3, 2)$$ is not.

FORMALLY:

A parking function of size $$n$$ is a sequence $$(a_1, \ldots, a_n)$$ of positive integers such that if $$b_1 \leq b_2 \leq \cdots \leq b_n$$ is the increasing rearrangement of $$a_1, \ldots, a_n$$, then $$b_i \leq i$$.

A parking function of size $$n$$ is a pair $$(L, D)$$ of two sequences $$L$$ and $$D$$ where $$L$$ is a permutation and $$D$$ is an area sequence of a Dyck path of size n such that $$D[i] \geq 0$$, $$D[i+1] \leq D[i]+1$$ and if $$D[i+1] = D[i]+1$$ then $$L[i+1] > L[i]$$.

The number of parking functions of size $$n$$ is equal to the number of rooted forests on $$n$$ vertices and is equal to $$(n+1)^{n-1}$$.

REFERENCES:

 [Beck] M. Beck, Stanford Math Circle - Parking Functions, October 2010, http://math.stanford.edu/circle/parkingBeck.pdf
 [Hag08] (1, 2, 3) The $$q,t$$ – Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Combinatorics of Macdonald Polynomials, James Haglund, University of Pennsylvania, Philadelphia – AMS, 2008, 167 pp.
 [Shin] (1, 2) H. Shin, Forests and Parking Functions, slides from talk September 24, 2008, http://www.emis.de/journals/SLC/wpapers/s61vortrag/shin.pdf
 [GXZ] (1, 2, 3) A. M. Garsia, G. Xin, M. Zabrocki, A three shuffle case of the compositional parking function conjecture, arXiv 1208.5796v1

AUTHORS:

• used non-decreasing_parking_functions code by Florent Hivert (2009 - 04)
• Dorota Mazur (2012 - 09)
sage.combinat.parking_functions.ParkingFunction(pf=None, labelling=None, area_sequence=None, labelled_dyck_word=None)

Return the combinatorial class of Parking Functions.

A parking function of size $$n$$ is a sequence $$(a_1, \ldots,a_n)$$ of positive integers such that if $$b_1 \leq b_2 \leq \cdots \leq b_n$$ is the increasing rearrangement of $$a_1, \ldots, a_n$$, then $$b_i \leq i$$.

A parking function of size $$n$$ is a pair $$(L, D)$$ of two sequences $$L$$ and $$D$$ where $$L$$ is a permutation and $$D$$ is an area sequence of a Dyck Path of size $$n$$ such that $$D[i] \geq 0$$, $$D[i+1] \leq D[i]+1$$ and if $$D[i+1] = D[i]+1$$ then $$L[i+1] > L[i]$$.

The number of parking functions of size $$n$$ is equal to the number of rooted forests on $$n$$ vertices and is equal to $$(n+1)^{n-1}$$.

INPUT:

• pf – (default: None) a list whose increasing rearrangement satisfies $$b_i \leq i$$
• labelling – (default: None) a labelling of the Dyck path
• area_sequence – (default: None) an area sequence of a Dyck path
• labelled_dyck_word – (default: None) a Dyck word with 1’s replaced by labelling

OUTPUT:

• A parking function

EXAMPLES:

sage: ParkingFunction([])
[]
sage: ParkingFunction()

sage: ParkingFunction()
Traceback (most recent call last):
...
ValueError:  is not a parking function.
sage: ParkingFunction([1,2])
[1, 2]
sage: ParkingFunction([1,1,2])
[1, 1, 2]
sage: ParkingFunction([1,4,1])
Traceback (most recent call last):
...
ValueError: [1, 4, 1] is not a parking function.
sage: ParkingFunction(labelling=[3,1,2], area_sequence=[0,0,1])
[2, 2, 1]
sage: ParkingFunction([2,2,1]).to_labelled_dyck_word()
[3, 0, 1, 2, 0, 0]
sage: ParkingFunction(labelled_dyck_word = [3,0,1,2,0,0])
[2, 2, 1]
sage: ParkingFunction(labelling=[3,1,2], area_sequence=[0,1,1])
Traceback (most recent call last):
...
ValueError: [3, 1, 2] is not a valid labeling of area sequence [0, 1, 1]
class sage.combinat.parking_functions.ParkingFunction_class(lst)
area()

Return the area of the labelled Dyck path corresponding to the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the sum of squares under and over the main diagonal the Dyck Path, corresponding to the parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.area()
6
sage: ParkingFunction([3,1,1,4]).area()
1
sage: ParkingFunction([4,1,1,1]).area()
3
sage: ParkingFunction([2,1,4,1]).area()
2
cars_permutation()

Return the sequence of cars that take parking spots 1 through $$n$$ and corresponding to the parking function.

For example, cars_permutation(PF) = [2, 4, 5, 6, 3, 1, 7] means that car 2 takes spots 1, car 4 takes spot 2, …, car 1 takes spot 6 and car 7 takes spot 7.

INPUT:

• self – parking function word

OUTPUT:

• the permutation of cars corresponding to the parking function and which is the same size as parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.cars_permutation()
[2, 4, 5, 6, 3, 1, 7]
sage: ParkingFunction([3,1,1,4]).cars_permutation()
[2, 3, 1, 4]
sage: ParkingFunction([4,1,1,1]).cars_permutation()
[2, 3, 4, 1]
sage: ParkingFunction([2,1,4,1]).cars_permutation()
[2, 1, 4, 3]
characteristic_quasisymmetric_function(q=None, R=Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field)

Return the characteristic quasisymmetric function of self.

The characteristic function of the Parking Function is the sum over all permutation labellings of the Dyck path $$q^{dinv(PF)} F_{ides(PF)}$$ where $$ides(PF)$$ (ides_composition()) is the descent composition of diagonal reading word of the parking function.

INPUT:

• q – (default: q = R('q')) a parameter for the generating function power
• R – (default: R = QQ['q','t'].fraction_field()) the base ring to do the calculations over

OUTPUT:

• an element of the quasisymmetric functions over the ring R

EXAMPLES:

sage: R = QQ['q','t'].fraction_field()
sage: (q,t) = R.gens()
sage: cqf = sum(t**PF.area()*PF.characteristic_quasisymmetric_function() for PF in ParkingFunctions(3)); cqf
(q^3+q^2*t+q*t^2+t^3+q*t)*F[1, 1, 1] + (q^2+q*t+t^2+q+t)*F[1, 2] + (q^2+q*t+t^2+q+t)*F[2, 1] + F
sage: s = SymmetricFunctions(R).s()
sage: s(cqf.to_symmetric_function())
(q^3+q^2*t+q*t^2+t^3+q*t)*s[1, 1, 1] + (q^2+q*t+t^2+q+t)*s[2, 1] + s
sage: s(cqf.to_symmetric_function()).nabla(power = -1)
s[1, 1, 1]
sage: p = ParkingFunction([3, 1, 2])
sage: p.characteristic_quasisymmetric_function()
q*F[2, 1]
sage: pf = ParkingFunction([1,2,7,2,1,2,3,2,1])
sage: pf.characteristic_quasisymmetric_function()
q^2*F[1, 1, 1, 2, 1, 3]
diagonal_composition()

Return the composition of the labelled Dyck path corresponding to the parking function.

For example, touch_composition(PF) = [4, 3] means that the first touch is four diagonal units from the starting point, and the second is three units further (see [GXZ] p. 2).

INPUT:

• self – parking function word

OUTPUT:

• the length between the corresponding touch points which of the labelled Dyck path that corresponds to the parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.touch_composition()
[4, 3]
sage: ParkingFunction([3,1,1,4]).touch_composition()
[2, 1, 1]
sage: ParkingFunction([4,1,1,1]).touch_composition()
[3, 1]
sage: ParkingFunction([2,1,4,1]).touch_composition()
[3, 1]

Return a diagonal word of the labelled Dyck path corresponding to parking function (see [Hag08] p. 75).

INPUT:

• self – parking function word

OUTPUT:

• returns a word, read diagonally from NE to SW of the pretty print of the labelled Dyck path that corresponds to self and the same size as self

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
[5, 1, 7, 4, 6, 3, 2]
sage: ParkingFunction([1, 1, 1]).diagonal_reading_word()
[3, 2, 1]
sage: ParkingFunction([1, 2, 3]).diagonal_reading_word()
[3, 2, 1]
sage: ParkingFunction([1, 1, 3, 4]).diagonal_reading_word()
[2, 4, 3, 1]
sage: ParkingFunction([1, 1, 1]).diagonal_word()
[3, 2, 1]
sage: ParkingFunction([1, 2, 3]).diagonal_word()
[3, 2, 1]
sage: ParkingFunction([1, 4, 3, 1]).diagonal_word()
[4, 2, 3, 1]
diagonal_word()

Return a diagonal word of the labelled Dyck path corresponding to parking function (see [Hag08] p. 75).

INPUT:

• self – parking function word

OUTPUT:

• returns a word, read diagonally from NE to SW of the pretty print of the labelled Dyck path that corresponds to self and the same size as self

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
[5, 1, 7, 4, 6, 3, 2]
sage: ParkingFunction([1, 1, 1]).diagonal_reading_word()
[3, 2, 1]
sage: ParkingFunction([1, 2, 3]).diagonal_reading_word()
[3, 2, 1]
sage: ParkingFunction([1, 1, 3, 4]).diagonal_reading_word()
[2, 4, 3, 1]
sage: ParkingFunction([1, 1, 1]).diagonal_word()
[3, 2, 1]
sage: ParkingFunction([1, 2, 3]).diagonal_word()
[3, 2, 1]
sage: ParkingFunction([1, 4, 3, 1]).diagonal_word()
[4, 2, 3, 1]
dinv()

Return the number of inversions of a labelled Dyck path corresponding to the parking function (see [Hag08] p. 74).

Same as the cardinality of dinversion_pairs().

INPUT:

• self – parking function word

OUTPUT:

• the number of dinversion pairs

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.dinv()
6
sage: ParkingFunction([3,1,1,4]).dinv()
3
sage: ParkingFunction([4,1,1,1]).dinv()
1
sage: ParkingFunction([2,1,4,1]).dinv()
2
dinversion_pairs()

Return the descent inversion pairs of a labelled Dyck path corresponding to the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the primary and secondary diversion pairs

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.dinversion_pairs()
[(0, 4), (1, 5), (2, 5), (1, 4), (2, 4), (3, 6)]
sage: ParkingFunction([3,1,1,4]).dinversion_pairs()
[(0, 3), (2, 3), (1, 2)]
sage: ParkingFunction([4,1,1,1]).dinversion_pairs()
[(1, 3)]
sage: ParkingFunction([2,1,4,1]).dinversion_pairs()
[(0, 3), (1, 3)]
ides()

Return the descents() sequence of the inverse of the diagonal_reading_word() of self.

Warning

Here we use the standard convention that descent labels start at $$1$$. This behaviour has been changed in trac ticket #20555.

For example, ides(PF) = [2, 3, 4, 6] means that descents are at the 2nd, 3rd, 4th and 6th positions in the inverse of the diagonal_reading_word() of the parking function (see [GXZ] p. 2).

INPUT:

• self – parking function word

OUTPUT:

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.ides()
[2, 3, 4, 6]
sage: ParkingFunction([3,1,1,4]).ides()

sage: ParkingFunction([4,1,1,1]).ides()
[2, 3]
sage: ParkingFunction([4,3,1,1]).ides()

ides_composition()

Return the descents_composition() of the inverse of the diagonal_reading_word() of corresponding parking function.

For example, ides_composition(PF) = [4, 2, 1] means that the descents of the inverse of the permutation diagonal_reading_word() of the parking function with word PF are at the 4th and 6th positions.

INPUT:

• self – parking function word

OUTPUT:

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.ides_composition()
[2, 1, 1, 2, 1]
sage: ParkingFunction([3,1,1,4]).ides_composition()
[2, 2]
sage: ParkingFunction([4,1,1,1]).ides_composition()
[2, 1, 1]
sage: ParkingFunction([4,3,1,1]).ides_composition()
[3, 1]
jump()

Return the sum of the differences between the parked and preferred parking spots.

See [Shin] p. 18.

INPUT:

• self – a parking function word

OUTPUT:

• the sum of the differences between the parked and preferred parking spots

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.jump()
6
sage: ParkingFunction([3,1,1,4]).jump()
1
sage: ParkingFunction([4,1,1,1]).jump()
3
sage: ParkingFunction([2,1,4,1]).jump()
2
jump_list()

Return the displacements of cars that corresponds to the parking function.

For example, jump_list(PF) = [0, 0, 0, 0, 1, 3, 2] means that car 1 through 4 parked in their preferred spots, car 5 had to park one spot farther (jumped or was displaced by one spot), car 6 had to jump 3 spots, and car 7 had to jump two spots.

INPUT:

• self – parking function word

OUTPUT:

• the displacements sequence of parked cars which corresponds to the parking function and which is the same size as parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.jump_list()
[0, 0, 0, 0, 1, 3, 2]
sage: ParkingFunction([3,1,1,4]).jump_list()
[0, 0, 1, 0]
sage: ParkingFunction([4,1,1,1]).jump_list()
[0, 0, 1, 2]
sage: ParkingFunction([2,1,4,1]).jump_list()
[0, 0, 0, 2]
luck()

Return the number of cars that parked in their preferred parking spots (see [Shin] p. 33).

INPUT:

• self – parking function word

OUTPUT:

• the number of cars that parked in their preferred parking spots

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.luck()
4
sage: ParkingFunction([3,1,1,4]).luck()
3
sage: ParkingFunction([4,1,1,1]).luck()
2
sage: ParkingFunction([2,1,4,1]).luck()
3
lucky_cars()

Return the cars that can park in their preferred spots. For example, lucky_cars(PF) = [1, 2, 7] means that cars 1, 2 and 7 parked in their preferred spots and all the other cars did not.

INPUT:

• self – parking function word

OUTPUT:

• the cars that can park in their preferred spots

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.lucky_cars()
[1, 2, 3, 4]
sage: ParkingFunction([3,1,1,4]).lucky_cars()
[1, 2, 4]
sage: ParkingFunction([4,1,1,1]).lucky_cars()
[1, 2]
sage: ParkingFunction([2,1,4,1]).lucky_cars()
[1, 2, 3]
parking_permutation()

Return the sequence of parking spots that are taken by cars 1 through $$n$$ and corresponding to the parking function.

For example, parking_permutation(PF) = [6, 1, 5, 2, 3, 4, 7] means that spot 6 is taken by car 1, spot 1 by car 2, spot 5 by car 3, spot 2 is taken by car 4, spot 3 is taken by car 5, spot 4 is taken by car 6 and spot 7 is taken by car 7.

INPUT:

• self – parking function word

OUTPUT:

• the permutation of parking spots that corresponds to the parking function and which is the same size as parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.parking_permutation()
[6, 1, 5, 2, 3, 4, 7]
sage: ParkingFunction([3,1,1,4]).parking_permutation()
[3, 1, 2, 4]
sage: ParkingFunction([4,1,1,1]).parking_permutation()
[4, 1, 2, 3]
sage: ParkingFunction([2,1,4,1]).parking_permutation()
[2, 1, 4, 3]
pretty_print(underpath=True)

Displays a parking function as a lattice path consisting of a Dyck path and a labelling with the labels displayed along the edges of the Dyck path.

INPUT:

• underpath – if the length of the parking function is less than or equal to 9 then display the labels under the path if underpath is True otherwise display them to the right of the path (default: True)

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.pretty_print()
___
_|1x
|7x  .
_____|3 . .
|5x x  . . .
_|4x  . . . .
|6x  . . . . .
|2 . . . . . .

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.pretty_print(underpath = false)
___
_| x  1
| x  . 7
_____|  . . 3
| x x  . . . 5
_| x  . . . . 4
| x  . . . . . 6
|  . . . . . . 2
sage: ParkingFunction([3, 1, 1, 4]).pretty_print()
_
_|4
___|1 .
|3x  . .
|2 . . .

sage: ParkingFunction([1,1,1]).pretty_print()
_____
|3x x
|2x  .
|1 . .

sage: ParkingFunction([4,1,1,1]).pretty_print()
_
_____|1
|4x x  .
|3x  . .
|2 . . .

sage: ParkingFunction([2,1,4,1]).pretty_print()
_
___|3
_|1x  .
|4x  . .
|2 . . .

sage: ParkingFunction([2,1,4,1]).pretty_print(underpath = false)
_
___|  3
_| x  . 1
| x  . . 4
|  . . . 2

sage: pf = ParkingFunction([1,2,3,7,3,2,1,2,3,2,1])
sage: pf.pretty_print()
_________
_______| x x x x   4
| x x x x x x x  .  9
| x x x x x x  . .  5
_| x x x x x  . . .  3
| x x x x x  . . . . 10
| x x x x  . . . . .  8
| x x x  . . . . . .  6
_| x x  . . . . . . .  2
| x x  . . . . . . . . 11
| x  . . . . . . . . .  7
|  . . . . . . . . . .  1
primary_dinversion_pairs()

Return the primary descent inversion pairs of a labelled Dyck path corresponding to the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the pairs $$(i, j)$$ such that $$i < j$$, and $$i^{th}$$ area = $$j^{th}$$ area, and $$i^{th}$$ label < $$j^{th}$$ label

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.primary_dinversion_pairs()
[(0, 4), (1, 5), (2, 5)]
sage: ParkingFunction([3,1,1,4]).primary_dinversion_pairs()
[(0, 3), (2, 3)]
sage: ParkingFunction([4,1,1,1]).primary_dinversion_pairs()
[]
sage: ParkingFunction([2,1,4,1]).primary_dinversion_pairs()
[(0, 3)]
secondary_dinversion_pairs()

Return the secondary descent inversion pairs of a labelled Dyck path corresponding to the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the pairs $$(i, j)$$ such that $$i < j$$, and $$i^{th}$$ area = $$j^{th}$$ area +1, and $$i^{th}$$ label > $$j^{th}$$ label

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.secondary_dinversion_pairs()
[(1, 4), (2, 4), (3, 6)]
sage: ParkingFunction([3,1,1,4]).secondary_dinversion_pairs()
[(1, 2)]
sage: ParkingFunction([4,1,1,1]).secondary_dinversion_pairs()
[(1, 3)]
sage: ParkingFunction([2,1,4,1]).secondary_dinversion_pairs()
[(1, 3)]
to_NonDecreasingParkingFunction()

Return the non-decreasing parking function which underlies the parking function.

INPUT:

• self – parking function word

OUTPUT:

• a sorted parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_NonDecreasingParkingFunction()
[1, 1, 2, 2, 5, 5, 6]
sage: ParkingFunction([3,1,1,4]).to_NonDecreasingParkingFunction()
[1, 1, 3, 4]
sage: ParkingFunction([4,1,1,1]).to_NonDecreasingParkingFunction()
[1, 1, 1, 4]
sage: ParkingFunction([2,1,4,1]).to_NonDecreasingParkingFunction()
[1, 1, 2, 4]
sage: ParkingFunction([4,1,2,1]).to_NonDecreasingParkingFunction()
[1, 1, 2, 4]
to_area_sequence()

Return the area sequence of the support Dyck path of the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the area sequence of the Dyck path

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_area_sequence()
[0, 1, 1, 2, 0, 1, 1]
sage: ParkingFunction([3,1,1,4]).to_area_sequence()
[0, 1, 0, 0]
sage: ParkingFunction([4,1,1,1]).to_area_sequence()
[0, 1, 2, 0]
sage: ParkingFunction([2,1,4,1]).to_area_sequence()
[0, 1, 1, 0]
to_dyck_word()

Return the support Dyck word of the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the Dyck word of the corresponding parking function

DyckWord()

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_dyck_word()
[1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0]
sage: ParkingFunction([3,1,1,4]).to_dyck_word()
[1, 1, 0, 0, 1, 0, 1, 0]
sage: ParkingFunction([4,1,1,1]).to_dyck_word()
[1, 1, 1, 0, 0, 0, 1, 0]
sage: ParkingFunction([2,1,4,1]).to_dyck_word()
[1, 1, 0, 1, 0, 0, 1, 0]
to_labelled_dyck_word()

Return the labelled Dyck word corresponding to the parking function.

This is a representation of the parking function as a list where the entries of 1 in the Dyck word are replaced with the corresponding label.

INPUT:

• self – parking function word

OUTPUT:

• the labelled Dyck word of the corresponding parking function which is twice the size of parking function word

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_labelled_dyck_word()
[2, 6, 0, 4, 5, 0, 0, 0, 3, 7, 0, 1, 0, 0]
sage: ParkingFunction([3,1,1,4]).to_labelled_dyck_word()
[2, 3, 0, 0, 1, 0, 4, 0]
sage: ParkingFunction([4,1,1,1]).to_labelled_dyck_word()
[2, 3, 4, 0, 0, 0, 1, 0]
sage: ParkingFunction([2,1,4,1]).to_labelled_dyck_word()
[2, 4, 0, 1, 0, 0, 3, 0]
to_labelling_area_sequence_pair()

Return a pair consisting of a labelling and an area sequence of a Dyck path which corresponds to the given parking function.

INPUT:

• self – the parking function word

OUTPUT:

• returns a pair (L, D) where L is a labelling and D is the area sequence of the underlying Dyck path

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_labelling_area_sequence_pair()
([2, 6, 4, 5, 3, 7, 1], [0, 1, 1, 2, 0, 1, 1])
sage: ParkingFunction([1, 1, 1]).to_labelling_area_sequence_pair()
([1, 2, 3], [0, 1, 2])
sage: ParkingFunction([1, 2, 3]).to_labelling_area_sequence_pair()
([1, 2, 3], [0, 0, 0])
sage: ParkingFunction([1, 1, 2]).to_labelling_area_sequence_pair()
([1, 2, 3], [0, 1, 1])
sage: ParkingFunction([1, 1, 3, 1]).to_labelling_area_sequence_pair()
([1, 2, 4, 3], [0, 1, 2, 1])
to_labelling_dyck_word_pair()

Return the pair (L, D) where L is a labelling and D is the Dyck word of the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the pair (L, D), where L is the labelling and D is the Dyck word of the parking function

DyckWord()

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_labelling_dyck_word_pair()
([2, 6, 4, 5, 3, 7, 1], [1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0])
sage: ParkingFunction([3,1,1,4]).to_labelling_dyck_word_pair()
([2, 3, 1, 4], [1, 1, 0, 0, 1, 0, 1, 0])
sage: ParkingFunction([4,1,1,1]).to_labelling_dyck_word_pair()
([2, 3, 4, 1], [1, 1, 1, 0, 0, 0, 1, 0])
sage: ParkingFunction([2,1,4,1]).to_labelling_dyck_word_pair()
([2, 4, 1, 3], [1, 1, 0, 1, 0, 0, 1, 0])
to_labelling_permutation()

Return the labelling of the support Dyck path of the parking function.

INPUT:

• self – parking function word

OUTPUT:

• the labelling of the Dyck path

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_labelling_permutation()
[2, 6, 4, 5, 3, 7, 1]
sage: ParkingFunction([3,1,1,4]).to_labelling_permutation()
[2, 3, 1, 4]
sage: ParkingFunction([4,1,1,1]).to_labelling_permutation()
[2, 3, 4, 1]
sage: ParkingFunction([2,1,4,1]).to_labelling_permutation()
[2, 4, 1, 3]
touch_composition()

Return the composition of the labelled Dyck path corresponding to the parking function.

For example, touch_composition(PF) = [4, 3] means that the first touch is four diagonal units from the starting point, and the second is three units further (see [GXZ] p. 2).

INPUT:

• self – parking function word

OUTPUT:

• the length between the corresponding touch points which of the labelled Dyck path that corresponds to the parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.touch_composition()
[4, 3]
sage: ParkingFunction([3,1,1,4]).touch_composition()
[2, 1, 1]
sage: ParkingFunction([4,1,1,1]).touch_composition()
[3, 1]
sage: ParkingFunction([2,1,4,1]).touch_composition()
[3, 1]
touch_points()

Return the sequence of touch points which corresponds to the labelled Dyck path after initial step.

For example, touch_points(PF) = [4, 7] means that after the initial step, the path touches the main diagonal at points $$(4, 4)$$ and $$(7, 7)$$.

INPUT:

• self – parking function word

OUTPUT:

• the sequence of touch points after the initial step of the labelled Dyck path that corresponds to the parking function

EXAMPLES:

sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.touch_points()
[4, 7]
sage: ParkingFunction([3,1,1,4]).touch_points()
[2, 3, 4]
sage: ParkingFunction([4,1,1,1]).touch_points()
[3, 4]
sage: ParkingFunction([2,1,4,1]).touch_points()
[3, 4]
sage.combinat.parking_functions.ParkingFunctions(n=None)

Return the combinatorial class of Parking Functions.

A parking function of size $$n$$ is a sequence $$(a_1, \ldots,a_n)$$ of positive integers such that if $$b_1 \leq b_2 \leq \cdots \leq b_n$$ is the increasing rearrangement of $$a_1, \ldots, a_n$$, then $$b_i \leq i$$.

A parking function of size $$n$$ is a pair $$(L, D)$$ of two sequences $$L$$ and $$D$$ where $$L$$ is a permutation and $$D$$ is an area sequence of a Dyck Path of size n such that $$D[i] \geq 0$$, $$D[i+1] \leq D[i]+1$$ and if $$D[i+1] = D[i]+1$$ then $$L[i+1] > L[i]$$.

The number of parking functions of size $$n$$ is equal to the number of rooted forests on $$n$$ vertices and is equal to $$(n+1)^{n-1}$$.

EXAMPLES:

Here are all parking functions of size 3:

sage: from sage.combinat.parking_functions import ParkingFunctions
sage: ParkingFunctions(3).list()
[[1, 1, 1], [1, 1, 2], [1, 2, 1], [2, 1, 1], [1, 1, 3], [1, 3, 1], [3, 1, 1],
[1, 2, 2], [2, 1, 2], [2, 2, 1], [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1],
[3, 1, 2], [3, 2, 1]]

If no size is specified, then ParkingFunctions returns the combinatorial class of all parking functions.

sage: PF = ParkingFunctions(); PF
Parking functions
sage: [] in PF
True
sage:  in PF
True
sage:  in PF
False
sage: [1,3,1] in PF
True
sage: [1,4,1] in PF
False

If the size $$n$$ is specified, then ParkingFunctions returns the combinatorial class of all parking functions of size $$n$$.

sage: PF = ParkingFunctions(0)
sage: PF.list()
[[]]
sage: PF = ParkingFunctions(1)
sage: PF.list()
[]
sage: PF = ParkingFunctions(3)
sage: PF.list()
[[1, 1, 1], [1, 1, 2], [1, 2, 1], [2, 1, 1], [1, 1, 3],
[1, 3, 1], [3, 1, 1], [1, 2, 2], [2, 1, 2], [2, 2, 1],
[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]
sage: PF3 = ParkingFunctions(3); PF3
Parking functions of size 3
sage: [] in PF3
False
sage:  in PF3
False
sage: [1,3,1] in PF3
True
sage: [1,4,1] in PF3
False
class sage.combinat.parking_functions.ParkingFunctions_all
class sage.combinat.parking_functions.ParkingFunctions_n(n)

The combinatorial class of parking functions of size $$n$$.

A parking function of size $$n$$ is a sequence $$(a_1, \ldots,a_n)$$ of positive integers such that if $$b_1 \leq b_2 \leq \cdots \leq b_n$$ is the increasing rearrangement of $$a_1, \ldots, a_n$$, then $$b_i \leq i$$.

A parking function of size $$n$$ is a pair $$(L, D)$$ of two sequences $$L$$ and $$D$$ where $$L$$ is a permutation and $$D$$ is an area sequence of a Dyck Path of size $$n$$ such that $$D[i] \geq 0$$, $$D[i+1] \leq D[i]+1$$ and if $$D[i+1] = D[i]+1$$ then $$L[i+1] > L[i]$$.

The number of parking functions of size $$n$$ is equal to the number of rooted forests on $$n$$ vertices and is equal to $$(n+1)^{n-1}$$.

EXAMPLES:

sage: PF = ParkingFunctions(3)
sage: PF.list()
[[1, 1, 1], [1, 1, 2], [1, 2, 1], [2, 1, 1], [1, 1, 3],
[1, 3, 1], [3, 1, 1], [1, 2, 2], [2, 1, 2], [2, 2, 1],
[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

sage: [ParkingFunctions(i).cardinality() for i in range(6)]
[1, 1, 3, 16, 125, 1296]

Warning

The precise order in which the parking function are generated or listed is not fixed, and may change in the future.

cardinality()

Return the number of parking functions of size n.

The cardinality is equal to $$(n+1)^{n-1}$$.

EXAMPLES:

sage: [ParkingFunctions(i).cardinality() for i in range(6)]
[1, 1, 3, 16, 125, 1296]
random_element()

Return a random parking function of size $$n$$.

The algorithm uses a circular parking space with $$n+1$$ spots. Then all $$n$$ cars can park and there remains one empty spot. Spots are then renumbered so that the empty spot is $$0$$.

The probability distribution is uniform on the set of $$(n+1)^{n-1}$$ parking functions of size $$n$$.

EXAMPLES:

sage: pf = ParkingFunctions(8)
sage: a = pf.random_element(); a  # random
[5, 7, 2, 4, 2, 5, 1, 3]
sage: a in pf
True
sage.combinat.parking_functions.from_labelled_dyck_word(LDW)

Return the parking function corresponding to the labelled Dyck word.

INPUT:

• LDW – labelled Dyck word

OUTPUT:

• the parking function corresponding to the labelled Dyck word that is half the size of LDW

EXAMPLES:

sage: from sage.combinat.parking_functions import from_labelled_dyck_word
sage: LDW = [2, 6, 0, 4, 5, 0, 0, 0, 3, 7, 0, 1, 0, 0]
sage: from_labelled_dyck_word(LDW)
[6, 1, 5, 2, 2, 1, 5]
sage: from_labelled_dyck_word([2, 3, 0, 0, 1, 0, 4, 0])
[3, 1, 1, 4]
sage: from_labelled_dyck_word([2, 3, 4, 0, 0, 0, 1, 0])
[4, 1, 1, 1]
sage: from_labelled_dyck_word([2, 4, 0, 1, 0, 0, 3, 0])
[2, 1, 4, 1]
sage.combinat.parking_functions.from_labelling_and_area_sequence(L, D)

Return the parking function corresponding to the labelling area sequence pair.

INPUT:

• L – a labelling permutation
• D – an area sequence for a Dyck word

OUTPUT:

• the parking function corresponding the labelling permutation L and D an area sequence of the corresponding Dyck path

EXAMPLES:

sage: from sage.combinat.parking_functions import from_labelling_and_area_sequence
sage: from_labelling_and_area_sequence([2, 6, 4, 5, 3, 7, 1], [0, 1, 1, 2, 0, 1, 1])
[6, 1, 5, 2, 2, 1, 5]
sage: from_labelling_and_area_sequence([1, 2, 3], [0, 1, 2])
[1, 1, 1]
sage: from_labelling_and_area_sequence([1, 2, 3], [0, 0, 0])
[1, 2, 3]
sage: from_labelling_and_area_sequence([1, 2, 3], [0, 1, 1])
[1, 1, 2]
sage: from_labelling_and_area_sequence([1, 2, 4, 3], [0, 1, 2, 1])
[1, 1, 3, 1]
sage.combinat.parking_functions.is_a(x, n=None)

Check whether a list is a parking function.

If a size $$n$$ is specified, checks if a list is a parking function of size $$n$$.