def ParkingFunction(pf=None, labelling=None, area_sequence=None,
labelled_dyck_word=None):
r"""
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([1])
[1]
sage: ParkingFunction([2])
Traceback (most recent call last):
...
ValueError: [2] 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]
"""
if pf is not None:
return ParkingFunction_class(pf)
elif labelling is not None:
if (area_sequence is None):
raise ValueError("must also provide area sequence along with labelling.")
if (len(area_sequence) != len(labelling)):
raise ValueError("%s must be the same size as the labelling %s" % (area_sequence, labelling))
if any(area_sequence[i] < area_sequence[i+1] and labelling[i] > labelling[i + 1] for i in range(len(labelling) - 1)):
raise ValueError("%s is not a valid labeling of area sequence %s" % (labelling, area_sequence))
return from_labelling_and_area_sequence(labelling, area_sequence)
elif labelled_dyck_word is not None:
return from_labelled_dyck_word(labelled_dyck_word)
elif area_sequence is not None:
DW = DyckWord(area_sequence)
return ParkingFunction(labelling=range(1, DW.size() + 1),
area_sequence=DW)
raise ValueError("did not manage to make this into a parking function")
def ParkingFunction(pf=None, labelling=None, area_sequence=None, labelled_dyck_word = None):
r"""
Returns 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([1])
[1]
sage: ParkingFunction([2])
Traceback (most recent call last):
...
ValueError: [2] 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]
"""
if pf is not None:
return ParkingFunction_class(pf)
elif labelling is not None:
if (area_sequence is None):
raise ValueError("must also provide area sequence along with labelling.")
if (len(area_sequence)!=len(labelling)):
raise ValueError("%s must be the same size as the labelling %s"%(area_sequence,labelling))
if any(area_sequence[i]<area_sequence[i+1] and labelling[i]>labelling[i+1] for i in range(len(labelling)-1)):
raise ValueError("%s is not a valid labeling of area sequence %s"%(labelling, area_sequence))
return from_labelling_and_area_sequence( labelling, area_sequence )
elif labelled_dyck_word is not None:
return from_labelled_dyck_word(labelled_dyck_word)
elif area_sequence is not None:
DW = DyckWord(area_sequence)
return ParkingFunction(labelling=range(1,DW.size()+1), area_sequence=DW)
else:
raise ValueError("did not manage to make this into a parking function")
