Symbolic method (combinatorics)

Summary

In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, Analytic Combinatorics, while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions.

During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of Bernoulli, Euler, Arthur Cayley, Schröder, Ramanujan, Riordan, Knuth, Comtet [fr], etc.). It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures translates, via some isomorphisms, into noteworthy identities on the corresponding generating functions. Following the works of Pólya, further advances were thus done in this spirit in the 1970s with generic uses of languages for specifying combinatorial classes and their generating functions, as found in works by Foata and Schützenberger[1] on permutations, Bender and Goldman on prefabs,[2] and Joyal on combinatorial species.[3]

Note that this symbolic method in enumeration is unrelated to "Blissard's symbolic method", which is just another old name for umbral calculus.

The symbolic method in combinatorics constitutes the first step of many analyses of combinatorial structures, which can then lead to fast computation schemes, to asymptotic properties and limit laws, to random generation, all of them being suitable to automatization via computer algebra.

Classes of combinatorial structures edit

Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration is the sum of the weights of the objects in the slots. We will first explain how to solve this problem in the labelled and the unlabelled case and use the solution to motivate the creation of classes of combinatorial structures.

The Pólya enumeration theorem solves this problem in the unlabelled case. Let f(z) be the ordinary generating function (OGF) of the objects, then the OGF of the configurations is given by the substituted cycle index

 

In the labelled case we use an exponential generating function (EGF) g(z) of the objects and apply the Labelled enumeration theorem, which says that the EGF of the configurations is given by

 

We are able to enumerate filled slot configurations using either PET in the unlabelled case or the labelled enumeration theorem in the labelled case. We now ask about the generating function of configurations obtained when there is more than one set of slots, with a permutation group acting on each. Clearly the orbits do not intersect and we may add the respective generating functions. Suppose, for example, that we want to enumerate unlabelled sequences of length two or three of some objects contained in a set X. There are two sets of slots, the first one containing two slots, and the second one, three slots. The group acting on the first set is  , and on the second slot,  . We represent this by the following formal power series in X:

 

where the term   is used to denote the set of orbits under G and  , which denotes in the obvious way the process of distributing the objects from X with repetition into the n slots. Similarly, consider the labelled problem of creating cycles of arbitrary length from a set of labelled objects X. This yields the following series of actions of cyclic groups:

 

Clearly we can assign meaning to any such power series of quotients (orbits) with respect to permutation groups, where we restrict the groups of degree n to the conjugacy classes   of the symmetric group  , which form a unique factorization domain. (The orbits with respect to two groups from the same conjugacy class are isomorphic.) This motivates the following definition.

A class   of combinatorial structures is a formal series

 

where   (the "A" is for "atoms") is the set of primes of the UFD   and  

In the following we will simplify our notation a bit and write e.g.

 

for the classes mentioned above.

The Flajolet–Sedgewick fundamental theorem edit

A theorem in the Flajolet–Sedgewick theory of symbolic combinatorics treats the enumeration problem of labelled and unlabelled combinatorial classes by means of the creation of symbolic operators that make it possible to translate equations involving combinatorial structures directly (and automatically) into equations in the generating functions of these structures.

Let   be a class of combinatorial structures. The OGF   of   where X has OGF   and the EGF   of   where X is labelled with EGF   are given by

 

and

 

In the labelled case we have the additional requirement that X not contain elements of size zero. It will sometimes prove convenient to add one to   to indicate the presence of one copy of the empty set. It is possible to assign meaning to both   (the most common example is the case of unlabelled sets) and   To prove the theorem simply apply PET (Pólya enumeration theorem) and the labelled enumeration theorem.

The power of this theorem lies in the fact that it makes it possible to construct operators on generating functions that represent combinatorial classes. A structural equation between combinatorial classes thus translates directly into an equation in the corresponding generating functions. Moreover, in the labelled case it is evident from the formula that we may replace   by the atom z and compute the resulting operator, which may then be applied to EGFs. We now proceed to construct the most important operators. The reader may wish to compare with the data on the cycle index page.

The sequence operator SEQ edit

This operator corresponds to the class

 

and represents sequences, i.e. the slots are not being permuted and there is exactly one empty sequence. We have

 

and

 

The cycle operator CYC edit

This operator corresponds to the class

 

i.e., cycles containing at least one object. We have

 

or

 

and

 

This operator, together with the set operator SET, and their restrictions to specific degrees are used to compute random permutation statistics. There are two useful restrictions of this operator, namely to even and odd cycles.

The labelled even cycle operator CYCeven is

 

which yields

 

This implies that the labelled odd cycle operator CYCodd

 

is given by

 

The multiset/set operator MSET/SET edit

The series is

 

i.e., the symmetric group is applied to the slots. This creates multisets in the unlabelled case and sets in the labelled case (there are no multisets in the labelled case because the labels distinguish multiple instances of the same object from the set being put into different slots). We include the empty set in both the labelled and the unlabelled case.

The unlabelled case is done using the function

 

so that

 

Evaluating   we obtain

 

For the labelled case we have

 

In the labelled case we denote the operator by SET, and in the unlabelled case, by MSET. This is because in the labeled case there are no multisets (the labels distinguish the constituents of a compound combinatorial class) whereas in the unlabeled case there are multisets and sets, with the latter being given by

 

Procedure edit

Typically, one starts with the neutral class  , containing a single object of size 0 (the neutral object, often denoted by  ), and one or more atomic classes  , each containing a single object of size 1. Next, set-theoretic relations involving various simple operations, such as disjoint unions, products, sets, sequences, and multisets define more complex classes in terms of the already defined classes. These relations may be recursive. The elegance of symbolic combinatorics lies in that the set theoretic, or symbolic, relations translate directly into algebraic relations involving the generating functions.

In this article, we will follow the convention of using script uppercase letters to denote combinatorial classes and the corresponding plain letters for the generating functions (so the class   has generating function  ).

There are two types of generating functions commonly used in symbolic combinatorics—ordinary generating functions, used for combinatorial classes of unlabelled objects, and exponential generating functions, used for classes of labelled objects.

It is trivial to show that the generating functions (either ordinary or exponential) for   and   are   and  , respectively. The disjoint union is also simple — for disjoint sets   and  ,   implies  . The relations corresponding to other operations depend on whether we are talking about labelled or unlabelled structures (and ordinary or exponential generating functions).

Combinatorial sum edit

The restriction of unions to disjoint unions is an important one; however, in the formal specification of symbolic combinatorics, it is too much trouble to keep track of which sets are disjoint. Instead, we make use of a construction that guarantees there is no intersection (be careful, however; this affects the semantics of the operation as well). In defining the combinatorial sum of two sets   and  , we mark members of each set with a distinct marker, for example   for members of   and   for members of  . The combinatorial sum is then:

 

This is the operation that formally corresponds to addition.

Unlabelled structures edit

With unlabelled structures, an ordinary generating function (OGF) is used. The OGF of a sequence   is defined as

 

Product edit

The product of two combinatorial classes   and   is specified by defining the size of an ordered pair as the sum of the sizes of the elements in the pair. Thus we have for   and  ,  . This should be a fairly intuitive definition. We now note that the number of elements in   of size n is

 

Using the definition of the OGF and some elementary algebra, we can show that

  implies  

Sequence edit

The sequence construction, denoted by   is defined as

 

In other words, a sequence is the neutral element, or an element of  , or an ordered pair, ordered triple, etc. This leads to the relation

 

Set edit

The set (or powerset) construction, denoted by   is defined as

 

which leads to the relation

 

where the expansion

 

was used to go from line 4 to line 5.

Multiset edit

The multiset construction, denoted   is a generalization of the set construction. In the set construction, each element can occur zero or one times. In a multiset, each element can appear an arbitrary number of times. Therefore,

 

This leads to the relation

 

where, similar to the above set construction, we expand  , swap the sums, and substitute for the OGF of  .

Other elementary constructions edit

Other important elementary constructions are:

  • the cycle construction ( ), like sequences except that cyclic rotations are not considered distinct
  • pointing ( ), in which each member of B is augmented by a neutral (zero size) pointer to one of its atoms
  • substitution ( ), in which each atom in a member of B is replaced by a member of C.

The derivations for these constructions are too complicated to show here. Here are the results:

Construction Generating function
    (where   is the Euler totient function)
   
   

Examples edit

Many combinatorial classes can be built using these elementary constructions. For example, the class of plane trees (that is, trees embedded in the plane, so that the order of the subtrees matters) is specified by the recursive relation

 

In other words, a tree is a root node of size 1 and a sequence of subtrees. This gives

 

we solve for G(z) by multiplying   to get

 

subtracting z and solving for G(z) using the quadratic formula gives

 

Another example (and a classic combinatorics problem) is integer partitions. First, define the class of positive integers  , where the size of each integer is its value:

 

The OGF of   is then

 

Now, define the set of partitions   as

 

The OGF of   is

 

Unfortunately, there is no closed form for  ; however, the OGF can be used to derive a recurrence relation, or using more advanced methods of analytic combinatorics, calculate the asymptotic behavior of the counting sequence.

Specification and specifiable classes edit

The elementary constructions mentioned above allow us to define the notion of specification. This specification allows us to use a set of recursive equations, with multiple combinatorial classes.

Formally, a specification for a set of combinatorial classes   is a set of   equations  , where   is an expression, whose atoms are   and the  's, and whose operators are the elementary constructions listed above.

A class of combinatorial structures is said to be constructible or specifiable when it admits a specification.

For example, the set of trees whose leaves' depth is even (respectively, odd) can be defined using the specification with two classes   and  . Those classes should satisfy the equation   and  .

Labelled structures edit

An object is weakly labelled if each of its atoms has a nonnegative integer label, and each of these labels is distinct. An object is (strongly or well) labelled, if furthermore, these labels comprise the consecutive integers  . Note: some combinatorial classes are best specified as labelled structures or unlabelled structures, but some readily admit both specifications. A good example of labelled structures is the class of labelled graphs.

With labelled structures, an exponential generating function (EGF) is used. The EGF of a sequence   is defined as

 

Product edit

For labelled structures, we must use a different definition for product than for unlabelled structures. In fact, if we simply used the cartesian product, the resulting structures would not even be well labelled. Instead, we use the so-called labelled product, denoted  

For a pair   and  , we wish to combine the two structures into a single structure. In order for the result to be well labelled, this requires some relabelling of the atoms in   and  . We will restrict our attention to relabellings that are consistent with the order of the original labels. Note that there are still multiple ways to do the relabelling; thus, each pair of members determines not a single member in the product, but a set of new members. The details of this construction are found on the page of the Labelled enumeration theorem.

To aid this development, let us define a function,  , that takes as its argument a (possibly weakly) labelled object   and relabels its atoms in an order-consistent way so that   is well labelled. We then define the labelled product for two objects   and   as

 

Finally, the labelled product of two classes   and   is

 

The EGF can be derived by noting that for objects of size   and  , there are   ways to do the relabelling. Therefore, the total number of objects of size   is

 

This binomial convolution relation for the terms is equivalent to multiplying the EGFs,

 

Sequence edit

The sequence construction   is defined similarly to the unlabelled case:

 

and again, as above,

 

Set edit

In labelled structures, a set of   elements corresponds to exactly   sequences. This is different from the unlabelled case, where some of the permutations may coincide. Thus for  , we have

 

Cycle edit

Cycles are also easier than in the unlabelled case. A cycle of length   corresponds to   distinct sequences. Thus for  , we have

 

Boxed product edit

In labelled structures, the min-boxed product   is a variation of the original product which requires the element of   in the product with the minimal label. Similarly, we can also define a max-boxed product, denoted by  , by the same manner. Then we have,

 

or equivalently,

 

Example edit

An increasing Cayley tree is a labelled non-plane and rooted tree whose labels along any branch stemming from the root form an increasing sequence. Then, let   be the class of such trees. The recursive specification is now  

Other elementary constructions edit

The operators CYCeven, CYCodd, SETeven, and SETodd represent cycles of even and odd length, and sets of even and odd cardinality.

Example edit

Stirling numbers of the second kind may be derived and analyzed using the structural decomposition

 

The decomposition

 

is used to study unsigned Stirling numbers of the first kind, and in the derivation of the statistics of random permutations. A detailed examination of the exponential generating functions associated to Stirling numbers within symbolic combinatorics may be found on the page on Stirling numbers and exponential generating functions in symbolic combinatorics.

See also edit

References edit

  1. ^ Foata, Dominique; Schützenberger, Marcel-P. (1970). "Théorie géométrique des polynômes Eulériens". Lectures Notes in Mathematics. Lecture Notes in Mathematics. 138. arXiv:math/0508232. doi:10.1007/BFb0060799. ISBN 978-3-540-04927-2.
  2. ^ Bender, Edward A.; Goldman, Jay R. (1971). "Enumerative uses of generating functions". Indiana University Mathematics Journal. 20 (8): 753–764. doi:10.1512/iumj.1971.20.20060.
  3. ^ Joyal, André (1981). "Une théorie combinatoire des séries formelles". Advances in Mathematics. 42: 1–82. doi:10.1016/0001-8708(81)90052-9.
  • François Bergeron, Gilbert Labelle, Pierre Leroux, Théorie des espèces et combinatoire des structures arborescentes, LaCIM, Montréal (1994). English version: Combinatorial Species and Tree-like Structures, Cambridge University Press (1998).
  • Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press (2009). (available online: http://algo.inria.fr/flajolet/Publications/book.pdf)
  • Micha Hofri, Analysis of Algorithms: Computational Methods and Mathematical Tools, Oxford University Press (1995).