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.structure.element.Element
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
- 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]
- grade()¶
Return the length of
self
.EXAMPLES:
sage: ndpf = NonDecreasingParkingFunctions(5) sage: len(ndpf.random_element()) 5
- to_dyck_word()¶
Implement 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]
- sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunctions(n=None)¶
Return the set 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 set 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 the set 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
- class sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunctions_all¶
Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
- graded_component(n)¶
Return the graded component.
EXAMPLES:
sage: P = NonDecreasingParkingFunctions() sage: P.graded_component(4) == NonDecreasingParkingFunctions(4) True
- class sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunctions_n(n)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
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
- Element¶
alias of
NonDecreasingParkingFunction
- cardinality()¶
Return 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
- one()¶
Return the unit of this monoid.
This is the non-decreasing parking function [1, 2, …, n].
EXAMPLES:
sage: ndpf = NonDecreasingParkingFunctions(5) sage: x = ndpf.random_element(); x # random sage: e = ndpf.one() sage: x == e*x == x*e True
- random_element()¶
Return a random parking function of the given size.
EXAMPLES:
sage: ndpf = NonDecreasingParkingFunctions(5) sage: x = ndpf.random_element(); x # random [1, 2, 2, 4, 5] sage: x in ndpf True
- 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 \(n\).