Non-Decreasing Parking Functions¶
A non-decreasing parking function of size \(n\) is a non-decreasing function \(f\) from \(\{1,\dots,n\}\) to itself such that for all \(i\), one has \(f(i) \leq i\).
The number of non-decreasing parking functions of size \(n\) is the \(n\)-th
Catalan number
.
The set of non-decreasing parking functions of size \(n\) is in bijection with
the set of Dyck words
of size \(n\).
AUTHORS:
- Florent Hivert (2009-04)
- Christian Stump (2012-11) added pretty printing
-
class
sage.combinat.non_decreasing_parking_function.
NonDecreasingParkingFunction
(lst)¶ Bases:
sage.combinat.combinat.CombinatorialObject
A non decreasing parking function of size \(n\) is a non-decreasing function \(f\) from \(\{1,\dots,n\}\) to itself such that for all \(i\), one has \(f(i) \leq i\).
EXAMPLES:
sage: NonDecreasingParkingFunction([]) [] sage: NonDecreasingParkingFunction([1]) [1] sage: NonDecreasingParkingFunction([2]) Traceback (most recent call last): ... ValueError: [2] is not a non-decreasing parking function sage: NonDecreasingParkingFunction([1,2]) [1, 2] sage: NonDecreasingParkingFunction([1,1,2]) [1, 1, 2] sage: NonDecreasingParkingFunction([1,1,4]) Traceback (most recent call last): ... ValueError: [1, 1, 4] is not a non-decreasing parking function
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classmethod
from_dyck_word
(dw)¶ Bijection from
Dyck words
. It is the inverse of the bijectionto_dyck_word()
. You can find there the mathematical definition.EXAMPLES:
sage: NonDecreasingParkingFunction.from_dyck_word([]) [] sage: NonDecreasingParkingFunction.from_dyck_word([1,0]) [1] sage: NonDecreasingParkingFunction.from_dyck_word([1,1,0,0]) [1, 1] sage: NonDecreasingParkingFunction.from_dyck_word([1,0,1,0]) [1, 2] sage: NonDecreasingParkingFunction.from_dyck_word([1,0,1,1,0,1,0,0,1,0]) [1, 2, 2, 3, 5]
TESTS:
sage: ndpf=NonDecreasingParkingFunctions(5); sage: list(ndpf) == [NonDecreasingParkingFunction.from_dyck_word(pf.to_dyck_word()) for pf in ndpf] True
-
to_dyck_word
()¶ Implements the bijection to
Dyck words
, which is defined as follows. Take a non decreasing parking function, say [1,1,2,4,5,5], and draw its graph:___ | . 5 _| . 5 ___| . . 4 _| . . . . 2 | . . . . . 1 | . . . . . 1
The corresponding Dyck word [1,1,0,1,0,0,1,0,1,1,0,0] is then read off from the sequence of horizontal and vertical steps. The converse bijection is
from_dyck_word()
.EXAMPLES:
sage: NonDecreasingParkingFunction([1,1,2,4,5,5]).to_dyck_word() [1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0] sage: NonDecreasingParkingFunction([]).to_dyck_word() [] sage: NonDecreasingParkingFunction([1,1,1]).to_dyck_word() [1, 1, 1, 0, 0, 0] sage: NonDecreasingParkingFunction([1,2,3]).to_dyck_word() [1, 0, 1, 0, 1, 0] sage: NonDecreasingParkingFunction([1,1,3,3,4,6,6]).to_dyck_word() [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0]
TESTS:
sage: ndpf=NonDecreasingParkingFunctions(5); sage: list(ndpf) == [pf.to_dyck_word().to_non_decreasing_parking_function() for pf in ndpf] True
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classmethod
-
sage.combinat.non_decreasing_parking_function.
NonDecreasingParkingFunctions
(n=None)¶ Returns the combinatorial class of Non-Decreasing Parking Functions.
A non-decreasing parking function of size \(n\) is a non-decreasing function \(f\) from \(\{1,\dots,n\}\) to itself such that for all \(i\), one has \(f(i) \leq i\).
EXAMPLES:
Here are all the-non decreasing parking functions of size 5:
sage: NonDecreasingParkingFunctions(3).list() [[1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 2, 3]]
If no size is specified, then NonDecreasingParkingFunctions returns the combinatorial class of all non-decreasing parking functions.
sage: PF = NonDecreasingParkingFunctions(); PF Non-decreasing parking functions sage: [] in PF True sage: [1] in PF True sage: [2] in PF False sage: [1,1,3] in PF True sage: [1,1,4] in PF False
If the size \(n\) is specified, then NonDecreasingParkingFunctions returns combinatorial class of all non-decreasing parking functions of size \(n\).
sage: PF = NonDecreasingParkingFunctions(0) sage: PF.list() [[]] sage: PF = NonDecreasingParkingFunctions(1) sage: PF.list() [[1]] sage: PF = NonDecreasingParkingFunctions(3) sage: PF.list() [[1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 2, 3]] sage: PF3 = NonDecreasingParkingFunctions(3); PF3 Non-decreasing parking functions of size 3 sage: [] in PF3 False sage: [1] in PF3 False sage: [1,1,3] in PF3 True sage: [1,1,4] in PF3 False
TESTS:
sage: PF = NonDecreasingParkingFunctions(5) sage: len(PF.list()) == PF.cardinality() True sage: NonDecreasingParkingFunctions("foo") Traceback (most recent call last): ... TypeError: unable to convert 'foo' to an integer
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class
sage.combinat.non_decreasing_parking_function.
NonDecreasingParkingFunctions_all
¶ Bases:
sage.combinat.combinat.InfiniteAbstractCombinatorialClass
TESTS:
sage: DW = NonDecreasingParkingFunctions() sage: DW == loads(dumps(DW)) True
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class
sage.combinat.non_decreasing_parking_function.
NonDecreasingParkingFunctions_n
(n)¶ Bases:
sage.combinat.combinat.CombinatorialClass
The combinatorial class of non-decreasing parking functions of size \(n\).
A non-decreasing parking function of size \(n\) is a non-decreasing function \(f\) from \(\{1,\dots,n\}\) to itself such that for all \(i\), one has \(f(i) \leq i\).
The number of non-decreasing parking functions of size \(n\) is the \(n\)-th Catalan number.
EXAMPLES:
sage: PF = NonDecreasingParkingFunctions(3) sage: PF.list() [[1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 2, 3]] sage: PF = NonDecreasingParkingFunctions(4) sage: PF.list() [[1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 1, 4], [1, 1, 2, 2], [1, 1, 2, 3], [1, 1, 2, 4], [1, 1, 3, 3], [1, 1, 3, 4], [1, 2, 2, 2], [1, 2, 2, 3], [1, 2, 2, 4], [1, 2, 3, 3], [1, 2, 3, 4]] sage: [ NonDecreasingParkingFunctions(i).cardinality() for i in range(10)] [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
Warning
The precise order in which the parking function are generated or listed is not fixed, and may change in the future.
AUTHORS:
- Florent Hivert
-
cardinality
()¶ Returns the number of non-decreasing parking functions of size \(n\). This number is the \(n\)-th
Catalan number
.EXAMPLES:
sage: PF = NonDecreasingParkingFunctions(0) sage: PF.cardinality() 1 sage: PF = NonDecreasingParkingFunctions(1) sage: PF.cardinality() 1 sage: PF = NonDecreasingParkingFunctions(3) sage: PF.cardinality() 5 sage: PF = NonDecreasingParkingFunctions(5) sage: PF.cardinality() 42
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sage.combinat.non_decreasing_parking_function.
is_a
(x, n=None)¶ Check whether a list is a non-decreasing parking function. If a size \(n\) is specified, checks if a list is a non-decreasing parking function of size \(n\).
TESTS:
sage: from sage.combinat.non_decreasing_parking_function import is_a sage: is_a([1,1,2]) True sage: is_a([1,1,4]) False sage: is_a([1,1,3], 3) True