# Combinatorial Classes

There are a lot of design issues which concern combinatorial classes. I'd like to discuss them here. The following are mostly notes from discussion with Nicolas. Once fixed, this material should end up in the doc of CombinatorialClass (I volunteer for this -Florent). Please add comments anywhere.

## Foreword (Teminology)

I need to fix some terminology. Maybe the name combinatorial class is bad in the context of object oriented programming. Should we call these "combinatorial set" ? Anyway, in the following when there is ambiguity I write OOClass and CClass.

## Documentation

We should agree on a overall structure of the main doc page of a CClass and write a template. I put here a stub for this. Before expanding it we should wait for the transition latex -> ReST.

```   r"""
My Favorite Combinatorial Class

...
"""```

## Interface and basic usage

The name of the OOClasses involved in building a CClass should be uniform : let's take the example of permutations

• the OOClass whose unique instance is the CClass of all permutations should be named Permutations (which is now permutations_all) the __init__ should take in charge to call sub-classes (see below discusion about factories) if needed.

• the OOClasses whose instances are sub-CClasses are called Permutations_constraints (eg:Permutations_n)

• the OOClass of the objects of a CClass are called by the singular form of the word as in Permutation

The following functions are standard and should be documented/publicized for all CClass:

• list()

• count() is the name fixed? coherent with the rest of sage? (+1 for count(). As far as I know it is standard. -jbandlow) It's not consistent with python count method :

```S.count(sub[, start[, end]]) -> int

Return the number of occurrences of substring sub in string
S[start:end].  Optional arguments start and end are
interpreted as in slice notation.```

I really prefer the length() which is used in words for example :).

• len is deprecated and should not be used

The following function should be written but are not supposed to be called directly by the user:

• __contains__ what should be the answer to [2,1,3] in permutations([2,1,3]) ?

• __iter__ this allows for the class itself to be iterated through. Therefore there is no need for iterator. iterator should be deprecated and removed. What should we do if there is several iterators ? See the discussion on the mailing list.

## Objects/Elements/Parents

The goal here is to be able to inherit smoothly from a combinatorial class to add extra mathematical structure (eg Poset, Group, Monoid).

• By default each element created by a combinatorial class should have a parent by default. It seems to me that this should be that largest combinatorial class which has no constraints (Permutations and not Permutations(4) for example). This should be simple to change this behavior.
• In case where we want to add more structure, We inherits from the combinatorial class. But the class of the element has to be changed. Therefore, elements should be constructed by an overloadable function and never by calling directly a class name. _element_constructor_ seems to be sage default.

• Let me give an example: I want to construct the symmetric group SG4 from the CClass Permutations(4) (what an original example :-). I clearly want to write SG4.count() and SG4.list(). A good way to do this is that the OOClass of SG4 (says SymGroup for short) inherits from the OOClass Permutations. But we have to be careful that SG4.list() construct instance of SG4.element_class (whatever it is) with parents SG4 and not instance of Permutation with parent Permutations().

A solution is to set two (lazy)_attribute in the combinatorial class named element_class and element_parent, and to define

```   def CClass._element_constructor_(self, x):
return self.element_class(z, self.element_parent)```

Of course for this to work we must ensure that any element constructed by the CClass is constructed through this function or by methods of self.element_class. Except as a default in self.element_class there should not be any reference in Permutations, Permutations_n and the others to the OOClass Permutation.

(I agree strongly with the first point. I don't understand the second point. Could you give an example or describe this more precisely? -jbandlow

Yes ! see up there -Florent)

## Bijections

When two combinatorial classes As and Bs with object a and b from OOclass A and B are in bijection, there are several possibilities

• a.to_B() and b.to_A() : this is practical for the user, from the combinatorial point of view. however there is no control on the class nor the parent of the results.

• As.from_B(b) and Bs.from_A(s) is more consistent. The combinatorial class knows its bijections.

• Hom(As, Bs).nameOftheBijection(a) ?

It is maybe not a very strong rule. Possible exception A is very standard and B is very exotic. Then maybe one can only write b.to_A() and B.from_A

(It might be cool to have some generic intelligence here. Suppose I add the new CClass C to sage and implement C._to(A,c). Then I would like it if when I call either C.to(A,c) or A.from(C,c), sage automatically tries both (if necessary) of A._from(C,c) and C._to(A,c). In other words, CombinatorialClass itself could have a method like:

```def to(self, class, element):
try:
return self._to(class, element)
except(NotImplementedError):
return class._from(self, element)```

And similar for from(). Thoughts? -jbandlow)

(I like this idea of generic intelligence which looks both at the domain and the image set. However, it it not clear for me if we prefer to write Cs.to(As,c) than C.to(A, c) (remember C and A is the OOClass of a and c whereas Cs and As are combinatorial classes. Bijection acts on objects but are beetween combinatorial classes. So I seems to by in favor of Cs.to(As,c) - Florent

• maybe we want to use coercion for that.
• How to deal with several bijections? extra arg Cs.to(As, c, 'name_of_the_bijection'). I maybe prefer Cs.to(As, 'name')(c)

• To avoid unuseful selection/dispatching logic I'd rather _to to try A._from_C and C._to_A.

## Combinatorial Class Factory

The goal here is to make it simple to make a subclass of a combinatorial class by adding some constraints. For example if p4=Permutations(4). The user may want to get the subclass of p4 of permutations of length say 5. So

• - Each combinatorial class should embed the necessary information to rebuild itself with extra constraints. - what is the syntax to add this extra constraints ?
• p4.subclass(length=5) ?
• other suggestion ? (I would prefer p4.with_constraint(length=5). I feel it is more clear than .subclass to a naive mathematician looking at a tab-completion list. -jbandlow)
- If no algorithm is known to handle the new constraints what should be the behavior ?
• raise an error suggesting the user to use a filter

• do the filter automatically (*Definitely* this. Why not? -jbandlow

Probably because we can't do that automatically. How do you choose from which class you filter in horrible things such as

`Permutations().with_constraint(descents=[3,5], shape=[4,3,1,1], length=7)`

As suggested by Nicolas, we can do that if there is a syntax for the user to tell what is the base class and what is the filter condition. - Florent)

(By the way, I *really* like the idea of Factories in general. -jbandlow)

Here is the copy paste of an old mail from Nicolas which was buried.

```> Factories of combinatorial classes:
>
> There are three levels:
>
> (a) Combinatorial class factories, like:
>     Permutations
>
> (b) Combinatorial classes, with list/count operations:
>     Permutations(4): models the combinatorial class of permutations of size 4
>     Permutations(4, descents = [2,1])
>     Permutations(4, greater_than = [3,1,2])
>     Permutations(bruhat_smaller=[3,1,2])
>     Permutations(left_bruhat_smaller=[3,1,2])
>     Permutations(4, left_bruhat_smaller=[3,1,2]): should raise an exception
>
>     Those are objects which may be an instance of many classes:
>      - Permutations_of_size
>      - Permutations_from_descents
>      - Permutations_lower_ideal_left_weak_order
>     Those classes are implementation details, and need not be known by the user
>
> (c) Data structures for combinatorial objects
>     class Permutation, class PermutationArray, class PermutationCycle
>
>     Those classes are mostly implementation details
>     They define the data structure and algorithms
>
> (d) Combinatorial objects:
>     Permutation([1,3,2])
>
>     One can always use a combinatorial class CC as constructor; in
>     that case, the result c is always an element of CC (i.e. c.parent() = CC)
>
>     Example:
>      - Permutation([1,2,3]) creates one permutation in Permutations()
>      - Permutations(4)([1,2,3,4]) also creates one permutation but this time in Permutations(4)
>
>     This might even be the recommended way.
>
> More involved example: tableaux
>
> (a) The Tableaux factory:
>     def Tableaux (# size options; only one can be specified?
>                   size=:, shape
>                   # content options; only one can be specified
>                 # standard = True if none is specified?
>                   standard, alphabet, evaluation, content
>                 young=True,
>                 constructor)
>
>     Maybe some aliases:
>      - def StandardYoungTableaux(*): Tableaux(*,standard=True, parent = StandardYoungTableaux())
>      - def SemiStandardYoungTableaux(*): Tableaux(*, parent = StandardYoungTableaux())
>
> (b) Tableaux(4):          standard young tableaux of size 4
>     Tableaux(shape=[4,1]) standard young tableaux of shape [4,1]
>
>     Tableaux(shape=[4,1], evaluation=[3,2])
>     Tableaux(shape=[4,1], alphabet=[1,2,4]) : semi standard young tableaux (obtained by crystal operations)
>
>     The default parent for the generated tableaux, depends on the young and standard options:
>      - Tableaux(standard = True, Young = True)
>      - Tableaux(standard=False, Young = True)
>      - Tableaux(standard=False, Young = True)
>
>
>
> (c) Data structures for combinatorial objects
>     Again, those classes are mostly implementation details
>     They define the data structure and algorithms (e.g. RSK will be in YoungTableau)
>
>     There may be more than one Tableau data structure (by rows, by
>     columns) in which case Tableau is a common abstract class.
>
> (d) Combinatorial objects:
>     Tableau([[2,3],[1,2]])
>     StandardTableau([[4,3],[1,2]])
>     StandardYoungTableau([[3,4],[1,2]])
>
>     One can always use a combinatorial class as constructor; this
>     might even be the recommended way:
>     Tableaux([[2,3],[1,2]])
>
>
> Different data structures and fundamental algorithms for combinatorial objects:
>  - Tableau(LabelledObject)
>    Indexation choice: t[row,col] with indexing 0 based and longest row first
>
>  - from Tableau derives:
>     class YoungTableau(Tableau):          implements e.g. RSK
>     class StandardTableau(Tableau)
>     class StandardYoungTableaux(YoungTableau,StandardTableau)
>
>  - SkewTableau(LabelledObject)       (LabelledObject)
>
>  - class AbstractTree(AbstractDigraph)
>    Defines precisely the syntax for constructors
>
>  - class MyTree(AbstractTree)
>
> ##############################################################################
> Factories, subfactories and subclasses
>
>
>  - Consider a factory and a combinatorial class C=Factory(constraints)
>    Then C.sub_class(extra_contraints) is the sub class of C satisfying
>    simultaneously the constraints for C (including the default ones)
>    and the extra_constraints. In practice, this will be constructed by
>    calling the factory with all the constraints set.
>
>    Consider for example the Tableaux factory (recall that the options
>    standard and evaluation are mutually incompatible)
>
>    Then, C = Tableaux() models the set of standard tableaux. It is
>    equivalent to C=Tableaux(standard=True).
>    Now, C.sub_class(evaluation=[3,2]) is *not* Tableaux(evaluation=[3,2])
>    but rather Tableaux(standard=True, evaluation=[3,2]) which should
>    trigger an error.
>
> ##############################################################################
> Factories as databases of algorithms
>
>    The Tableaux factory will typically be implemented using a database like:
>     {
>       { standard=True, n:     "*"} : StandardTableauxBySize
>       { standard=True, shape: "*"} : StandardTableauxByShape
>       { alphabet: "*", shape: "*"} : SemiStandardTableauxByShapeAndAlphabetCrystal
>     }
>
>    This allows for two nice features
>     - A posteriori extensions of the database by pluging in new algorithm
>     - Construction of partially specialized subfactory with
>       Tableaux.subfactory(shape="*", alphabet="*")
>       to bypass the option checking for those case speed is at a premium
>
>    Questions:
>
>     - in some cases, we may want to split between the different cases
>       differently for counting and listing. Should we support this?
>     - does it make sense to define StandardTableaux as
>       StandardTableaux = Tableaux.sub_factory(standard=True)
>
>
>
>
> CC = CombinatorialClass
>
> To create the combinatorial classes:
>
> for x in Tableaux(3):
>     ...
>
> Comp = Compositions()
> Compositions(4)              : CC of elements of Compositions()
> Compositions(4, parent=Compositions(4)) : CC of elements of Compositions(4)
>
> Trees(4, constructor=MyTree) : CC of instances of my_data_structure
>
> Comp.sub_class(4)            : returns Compositions(4)
>
> Comp([3,1,2])                : returns the element [3,1,2] of Comp
>
> Arrangements()
>
>
>
> Permutations(4)([1,3,2,4]) : returns an element of Permutations(4)
>
> In general if C is a combinatorial class C(...) returns an element of C
> (Example C=Trees(4))```