Equivalence classes of nodes in trees and rational generating functions
Let denote the number of nodes at a distance from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence is described. This criterion is applied to counting the number of conjugacy classes of commuting tuples in finite groups and the number of isomorphism classes of representations of polynomial algebras over finite fields. The method for computing the rational generating functions, when applied to the study of point configurations in finite sets, gives rise to some classical combinatorial results on Bell numbers and Stirling numbers of the second kind. When applied to the study of vector configurations in a finite vector space, it reveals a connection between counting such configurations and Gaussian binomial coefficients.
Key words and phrases:Trees, generating functions, Bell numbers, Stirling numbers, Gaussian binomial coefficients, simultaneous conjugacy classes, commuting tuples, modules for polynomial algebras, finite fields
2010 Mathematics Subject Classification:05A15
This paper begins by describing a technique for proving the rationality of, and often explicitly computing, ordinary generating functions of certain combinatorial sequences (Theorem 2). It applies to sequences whose th term can be expressed as a the number of nodes in a rooted tree at a distance from the root. The rationality rests upon the finiteness of the number of what are called lineal isomorphism classes of nodes.
The counting of simultaneous conjugacy classes of commuting -tuples in a finite group is, in general, a difficult combinatorial problem. However, the rationality of the generating function associated to this count turns out to be an easy consequence of Theorem 2 (see Theorem 3). These generating functions are computed explicitly for the first five symmetric groups (Table 2).
A slight variant of this result also shows that if denotes the number of isomorphism classes of -dimensional representations of the polynomial algebra (here is a finite field of order ), then (as a sequence in ) has a rational generating function (Theorem 6).
The method from Theorem 2 for computing generating functions can sometimes be applied advantageously even to situations where rationality is easy to see by other methods. For instance, when applied to counting point configurations in finite sets, it leads to beautiful classical results concerning Bell numbers and Stirling numbers of the second kind. When applied to counting vector configurations in finite vector spaces, it leads to the discovery of a new interpretation of Gaussian binomial coefficients.
2. Lineal equivalence and rational generating functions
Let denote the vertex set of a rooted tree with root vertex . Let denote the set of vertices of which are a distance from . Then is a disjoint union:
We will give a sufficient condition for the formal generating function
to be a rational function in and a technique for its computation.
If and are connected by an edge, then we say that is a child of , and write . More generally, if and for some are such that there exists a sequence
then we say that is a descendant of (under our definition is a descendant of ).
For each , let denote the full subtree consisting of the descendants of . This is again a rooted tree, with root .
Definition (Lineal Isomorphism).
Two vertices and of are said to be lineally isomorphic if the rooted trees and are isomorphic (in other words, there is a graph isomorphism taking to ).
Clearly, lineal isomorphism is an equivalence relation on . The equivalence classes of this relation are called lineal isomorphism classes.
If and are lineally isomorphic nodes in a rooted tree , then for any lineal isomorphism class , the number of children of in is equal to the number of children of in .
Since and are lineally isomorphic, there exists an isomorphism of rooted trees. This isomorphism defines a bijection from the children of in to the children of in . ∎
Let be a rooted tree with finitely many lineal isomorphism classes , the root of of lying in . Let be the matrix where is the number of children that a node in the class has in the class . Then, for each ,
Here, for each , denotes the th coordinate vector, viewed as an matrix, and its transpose. In particular, the sequence has a rational generating function for each . Consequently,
Here is the all-ones row vector.
Let for and . Let denote the column vector with coordinates . The hypothesis that the root of lies in implies that .
Since each node in contributes elements to ,
The recurrence relation (4) can be written in matrix form as:
upon iterating which (and using the fact that ), we get
Figure 1 shows the vertices within distance from the root of a tree with two lineal equivalence classes, labelled ’’ and ’’. The vertices of type have three children, one of type and two of type , while the vertices of type have no children of type and two children of type .
The “branching matrix” is
Therefore, if and are the numbers of nodes of type and respectively which lie at distance from the root, then
If is the set of nodes at a distance from the root, we have
3. Conjugacy classes of commuting tuples in groups
Let be a finite group. Then acts on for each non-negative integer by simultaneous conjugacy:
If is the number of orbits for the action of in , it is not difficult to see that is a rational function in . Indeed, by Burnside’s lemma
where denotes the centralizer of in . Therefore,
which is a rational function in .
Taking (the symmetric group on symbols),
Here signifies that is a partition of , and for each such partition, denotes the cardinality of the centralizer in of a permutation with cycle type . If, for each positive integer , is the number of occurrences of in , then
The values of for small values of are given in Table 1.
A more subtle problem is that of counting simultaneous conjugacy classes of -tuples of commuting elements in . For each , let
In particular, is the trivial group.
Let denote the number of orbits for the action of on by simultaneous conjugation, as given in (5). Consider the generating function
Because the elements are no longer independent, Burnside’s lemma can no longer be used to prove the rationality of . However, Theorem 2 allows us to show that is rational in , and gives an algorithm to compute it for any finite group.
For every finite group , the formal power series defined above is a rational function of .
Let denote the set of -orbits in . Say that is connected to by an edge if there exists such that . This gives the structure of a rooted tree with root being the unique element of . For , let
We will see in Theorem 4 below that the -orbit of is lineally isomorphic to the -orbit of in if the group is isomorphic to the group . Since each of these centralizers is a subgroup of the finite group , there are only finitely many possible isomorphism classes for them, and so only finitely many lineal isomorphism classes in . Thus Theorem 2 applies, and is a rational function of . ∎
Suppose that lies in the -orbit , then the full subtree of (rooted at ) consisting of descendants of in is isomorphic to the rooted tree associated to the group .
Let . Define a map by
It is easy to check that this map induces an isomorphism of rooted trees . ∎
We now consider the examples of symmetric groups: since is abelian, , which was computed earlier in this section. Now has three conjugacy classes, which lie in different lineal isomorphism classes in since they have non-isomorphic centralizers. The centralizers are given in the following table:
With the exception of the class of the identity element (with cycle type ) each of these centralizers is abelian. If the orbit of a tuple has abelian centralizer, then all its descendants are lineally isomorphic to it. For the singleton orbit of the identity element, we once again have three children, one corresponding to each partition of . Thus every pair of commuting elements on has centralizer isomorphic to that of an element of . The branching matrix of Theorem 2 is
A routine calculation shows that
The group five conjugacy classes with centralizers given by:
The only troublesome case here is . The centralizer group in this case is a non-abelian group of order , which we now proceed to analyse: for concreteness, consider the centralizer of the permutation . The centralizer subgroup consists of the permutations:
This group has five conjugacy classes, with centralizers given by:
Thus, as in the case of , the centralizers of pairs are all centralizers of elements in . The branching matrix is
Similarly for we have
The classes corresponding to the partitions and can be clubbed, as they have isomorphic centralizers (and therefore are lineally isomorphic). In , we have the following count of classes and their centralizers:
The only centralizer here which is not the centralizer of an element of is , which is abelian. The branching matrix is given by:
whence one may compute:
Our calculations of for small values of are summarized in Table 2.
The techniques at hand are not strong enough to derive an analog of the formula in Example 2 for for general .
Theorem 3 can also be stated for finite algebras:
Let be a finite ring, its multiplicative group of units, and let
Then acts on by simultaneous similarity:
Let denote the number of orbits for the action of on . Then the generating function
is a rational function of .
The proof is similar to Theorem 3: Let
where denotes the subring of elements of that commute with . The -orbits of and are lineally isomorphic if the rings and are isomorphic. ∎
Let be a finite field of order . Taking to be the algebra of matrices with entries in in Theorem 5 gives simultaneous similarity classes of commuting -tuples in . An -tuple of commuting matrices is nothing but an -module. Two modules are isomorphic if and only if the corresponding -tuples are simultaneously similar. So we have:
Let denote the finite field with elements, and for each positive integer let denote the number of isomorphism classes of -dimensional modules for the polynomial algebra . Then the generating function:
is a rational function of .
The polynomials are quite difficult to compute for , but seem to have very interesting combinatorial properties, an investigation of which is the subject of . For example,
The details of these (and further) calculations can be found in .
As with groups, counting of -orbits in (instead of ) is much easier, because of the applicability of Burnside’s lemma. When this becomes the problem of counting isomorphism classes of -dimensional representations of the free algebra , which in turn is a special case of the problem of counting representations of a quiver with the fixed dimension vector, a well-developed program which was started by Kac  in 1983 and culminated in the recent work of Hausel, Letellier and Rodriguez-Villegas  and Mozgovoy . In contrast, we do not even know that is a polynomial in for .
The counting of isomorphism classes of -modules appears to be related to the counting of similarity classes of matrices in finite quotients of discrete valuation rings. Let be a discrete valuation ring with residue field . Let denote the maximal ideal of . Matrices are said to be similar if for some . From the work of Singla , Jambor and Plesken  and Prasad, Singla and Spallone , we know that the number of isomorphism classes of -dimensional -modules (what we have called above) is equal to the number of similarity classes in . Further, by comparing the values for quoted above with the results obtained by Avni, Onn, Prasad and Vaserstein  we find that for , matches the number of similarity classes in for and all .
One is led to the following conjecture:
The number of similarity classes in is equal to the number of isomorphism classes of -dimensional -modules for all positive integers and .
4. Point and vector configurations
The symmetric group acts on the set . An -point configuration in is, by definition, an orbit of for its action on by
For example, there are two -point configurations in for : either the points and coincide, or they are distinct. Likewise, there are five -point configurations in for , represented by
Let denote the number of -point configurations in .
We may compute using Burnside’s lemma. With the notation of Example 2,
However, we shall see below that using Theorem 2 for the same computation leads to the standard ordinary generating functions and recurrence relations for Bell numbers [13, A000110] and Stirling numbers of the second kind [13, A008277].
Let denote the set of orbits in , and . Say that is a child of if there exists such that . We say that has type if, for any , the number of distinct elements in the set is . Clearly, if has type , then each of its children has type either or . Also, and lie in the same -orbit if and only if there exists a permutation which fixes and maps to .
Now suppose that is of type and . If, for some , the orbit of is also of type , then coincides with one of and is therefore fixed by any which fixes them. Thus, a node of type has children of type . On the other hand, if the orbit of is of type , then is different from each of and can therefore be permuted to any other element of that is distinct from while fixing them. It follows that a node of type has child of type .
The branching matrix is given by
a matrix whose diagonal entries are , with ’s just below the diagonal and with all other entries zero. One easily computes
The sequence has generating function
Each -tuple gives rise to an equivalence relation on ; indices are equivalent if . Two tuples are in the same -orbit if and only if they give rise to the same equivalence relation on . Thus the number of -point configurations in is nothing but the number of equivalence relations on with at most equivalence classes. For , this number is the well-known Bell number . Under the correspondence between -point configurations in and equivalence relations on with at most equivalence classes, point configurations of type map to equivalence relations with exactly equivalence classes. The number of equivalence relations with exactly equivalence classes is the well-known Stirling number of the second kind, usually denoted or [14, Section 1.9]. The identity (7) becomes a well-known generating function for Stirling numbers of the second kind [14, Eq. (1.94c)]:
By definition, the Bell number is the number of equivalence relations on a set of order . Clearly, which equals provided that . Thus the ordinary generating function for Bell numbers (see, e.g., [8, Lemma 8]) is obtained:
Let denote a finite field with elements. The general linear group acts on the vector space , and therefore also on -tuples of vectors in it:
for and for . A configuration of vectors in is an orbit of on .
Let denote the set of -orbits in . Say that is a child of if there exists such that . Let . We say that has type if, for any , the dimension of the subspace spanned by the set is . If is of type , then a child of must be of type or . If is of type , then lies in the span of . Therefore any element of that fixes fixes as well. Therefore, a tuple of type has children of type . If, on the other hand, and both have type , then and are linearly independent of , so there exists mapping to while fixing . Therefore, a tuple of type has only one child of type .
The branching matrix is given by
The generating function fot is
The quantity does not depend on so long as . The stable value of this quantity may be regarded as an analog of the Bell number for which we get the ordinary generating function:
Likewise, if we only count those -vector configurations in which span an -dimensional subspace of