arXiv search

I check the links on my web page from time to time. Yesterday, I checked the links to my arXiv papers, and was a bit surprised to get a message that “this search method is deprecated”.

It took a little bit of research and experimentation to figure out how to use their new search function. One change: before, I had separate searches for my papers in mathematics, statistics and computer science, because I didn’t know how to combine them; now I have a single unified list (partly because I don’t know how to separate it, but also because this seems to me to be better; the distinctions were somewhat artificial).

Anyway, I put a link on the right-hand bar of this blog. It should work OK; let me know if there are problems.

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Polynomially bounded orbit counts

The best news I had yesterday was an email from Justine Falque with a link to a paper that she and Nicolas Thiéry have just put on the arXiv. The 12-page document is only the “short version”, and a longer paper with complete proofs will appear at some point. Nevertheless, there is enough here to persuade me that they have solved the problem (open since the 1970s) with appropriate tools, and have a very nice result.

A permutation group G, acting on the infinite set X, is said to be oligomorphic if the number of orbits of G on the set of n-element subsets of X is finite for all natural numbers n.

The first case to think about is that where the number of orbits is 1 for all n, in other words, the group is highly homogeneous (or highly set-transitive). Such groups were the subject of my first theorem about infinite permutation groups, answering a question posed by John McDermott. Either such a group preserves or reverses a linear or circular order on X, or it is transitive on ordered n-tuples for all n (that is, it is highly transitive). In fact, there is no need to worry about the complexities of uncountable sets or proper subgroups with the same degree of transitivity: the downward Löwenheim–Skolem theorem guarantees that anything that can happen happens on a countable set, and by taking the closure in the topology of pointwise convergence we need only deal with five very specific groups.

The next step might be to investigate the case where the number of orbits has polynomial growth, that is, is bounded by a polynomial in n. To simplify matters, let us assume that G acts transitively on X. At the time, I was aware that certain imprimitive groups (groups preserving partitions of X) would give examples. For example, take the group preserving all partitions of X into k infinite blocks. The orbit of an n-set is determined by the sizes of its intersections with these blocks, and so the number of orbits is the number of partitions of n into at most k parts; this grows like a polynomial of degree k−1. Also, the group preserving a partition of X into parts of size k has the number of orbits on n-sets equal to the number of partitions of n into parts of size at most k. Duality of partitions (corresponding to transposing the Ferrers diagram) shows that these numbers are equal.

The summer before Dugald Macpherson started work on his DPhil under my supervision, I suggested that he might like to consider whether there could be a primitive group with polynomial growth. This was motivated by a construction I had devised. We can describe the orbit structure on finite sets for an oligomorphic group by a graded algebra. Let Vn be the vector space of G-invariant functions from the set of n-element subsets of X to our favourite field of characteristic zero. The dimension of Vn is equal to the number of orbits on n-sets. We can make the direct sum of these spaces into a graded algebra as follows. Given f in Vn and g in Vm, define fg to be the function in Vn+m as follows: Let L be an (n+m)-set. To evaluate (fg)(L), for each n-subset K of L, we multiply f(K) by g(L\K); then sum over all choices of K. This algebra is commutative and associative, and has an identity. If it were finitely generated, then the dimensions of the homogeneous components Vn would be bounded by a polynomial in n.

By the end of the summer, Dugald had solved this problem: If G is primitive, then either it is highly homogeneous, or the orbit counts grow at least fractional-exponentially (like the unrestricted partition function). Subsequently he improved the growth rate in the second case to straight exponential.

What was left open was a precise description of the growth in the polynomially bounded case. This is what Falque and Thiéry have now produced. In a beautiful piece of work, they have shown that, if the orbit numbers are polynomially bounded, then the graded algebra constructed above is Cohen–Macaulay, a strong ring-theoretic property which implies that the Hilbert series (the generating function for the orbit numbers) has a very simple form: it is a rational function whose numerator is a polynomial with non-negative integer coefficients and whose denominator is a product of factors of the form (1−xa) for various a.

Falque and Thiéry remark that this is just a step towards a complete classification of permutation groups with polynomial growth for the orbit counts. But certainly an important step.

So the behaviour of the two examples I gave earlier, where the Hilbert series is 1/((1−x)(1−x2)…(1−xk)), is not just a special case, but is typical of the general situation.

So what next? Here are two questions which occur to me.

  1. If G is transitive and oligomorphic, then any chain of blocks of imprimitivity is finite, and in the polynomial growth case there is at most one infinite “gap” in such a chain (since, if there were two, we could encode partitions of integers). What if we allow two infinite gaps: is it possible to describe what happens? The natural restriction on growth rates would be just below exponential (the growth for the iterated wreath product of three symmetric groups is faster than any fractional exponential but below straight exponential.)
  2. A big open question which will probably require quite different techniques: what can be said about the spectrum of exponential constants for growth rates of primitive groups? The best lower bound is about 1.324 by Francesca Merola; but very little else is known!
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Mathematical collaboration

I have just spent an entire weekend at a workshop on mathematical collaboration. It blew a huge hole in the time that I had to get on with urgent work that needs to be done, but it was such a good experience that I don’t really mind. I had been thinking of titling this post “I’m a mathematician – get me out of here!”, but in fact the mathematicians were not treated just as performing seals or experimental rats but were made to feel like collaborators in the enterprise.

The meeting was in the philosophy department, which stands on the edge of the cliff with a stunning view over the sea, the West Sands, and beyond. I am ashamed to say that after five years in St Andrews, this was the first time I had been into the building, though I had been told one of its features. It has two doors, labelled (if I recall correctly) Metaphysics and Moral Philosophy, but you can enter either, they lead into the same space.

The meeting was organised by Fenner Tanswell (St Andrews) and Josh Habgood-Coote (now in Bristol), and was associated in some way with Ursula Martin’s grant. Ursula showed up on both days, but was visibly coming down with some illness (possibly related to the ones that struck me down three times this winter) and didn’t stay long.

The first talk was by Josh, who spoke about authorship, a somewhat vexed question in these days of REF. He began by showing us a 12-page physical paper (on a measurement of the mass of the Higgs boson) which had 5135 authors, whose names covered 16 pages. His proposal was to split the functions of authorship, so that a paper has contributors (who did the work), a spokesperson (responsible for the write-up), and a guarantor (who takes responsibility for the project). The mathematicians in the audience were not entirely happy with this. Our default position is that every author is a guarantor; we expect every author to have read and approved all of the paper. But clearly not all is well in the academy. There are guest authors (the head of the institution whose name must be put on all papers) and ghost authors (a problem in the pharmaceutical world, and also in economics, where the research is done and the paper written entirely in the commercial organisation and an academic author is added just to make it look respectable). Read Ben Goldacre on this!

Next was Benedikt Loewe, who teaches in Amsterdam, Hamburg, and Cambridge. He described a course he put on for potential researchers in the “philosophy of mathematical practice”. Here we first heard the term “experimental philosophy”. He used the analogy of buying clothes. If you want clothes, you probably go to a shop. But perhaps nothing in the shop fits very well. You might consider a bespoke tailor. But this is more expensive, and the tailor might not make exactly what you want because of some kind of artistic pride. So as a last resort, you might try to make your own clothes, knowing that they may end up not terribly well made. In the same way, “experimental philosophy” tries to use the services of neuropsychologists (for example), but many end up doing the experiments themselves. Philosophy of scientific practice has existed for some years, but these people regard mathematics with “a mixture of awe and lack of interest”.

Kamilla Rekvenyi, a mathematics undergraduate in St Andrews, presented material which had grown out of her History of Mathematics project, on “Paul Erdős’ mathematics as a social activity”. She told about how he encouraged young mathematicians by asking interesting questions (exemplified by letters he wrote to the current Regius Professor of Mathematics in St Andrews, who was in the audience), and about the row between Erdős and Atle Selberg about the elementary proof of the Prime Number Theorem (where one could say that Erdős thought they were collaborating but Selberg didn’t).

I was very impressed by Stephen Crowley, the last speaker of the day. His topic was “Does collaboration make mathematicians virtuous?” His answer was “no”; but since he so clearly had come to learn, and since he himself showed so clearly one of the virtues he mentioned (humility – I hope he doesn’t mind me saying this), I was not entirely convinced. He made a good argument that mathematics is an ideal test case for questions of this sort. In any case, this depends on something called “virtue theory” which several people mentioned, but which I have been unable to get a clear grasp of. Stephen managed to confuse us by wearing a T-shirt with a map of Idaho labelled “Iowa”; whether this was a subtle dig at the standard of geographic knowledge of many Americans, or something quite different, I was not really sure.

Then there was a panel discussion, which I won’t even attempt to capture, and dinner, where we were given a good example of the opposite of virtue by a large party of golfers at the next table who drank lots of beer and spoke so loudly that conversation on our table was severely curtailed.

The next day began with a joint presentation by Fenner and Colin Rittberg on “Epistemic injustice in mathematics”. One of his examples concerned a mathematician who claimed to have been unjustly treated because her papers were rejected by journals with the comment that “that is not difficult to prove”, even though no published proof could be cited. There is a real issue here, but the mathematicians were confused by the fact that the authors called such results “folk theorems”, more or less defining this term to mean results which “the experts” know even though there is no proof in the literature. My understanding, confirmed by others present, is that a “folk theorem” is one where we do not know who first proved it; if we use it, we have either to refer to a textbook proof or to give our own proof, so there are often several proofs in the literature of such results (perhaps also a problem but hardly an injustice). We suggested that another term be used for the first concept, possibly “ghost theorem” – suggestions for better terminology welcome!

Chris Kelp talked about “Inquiry, Knowledge and Understanding”. This was a more traditional philosophy talk. He believes that the aim of an inquiry is to gain knowledge, and the inquiry is successful if the knowledge is gained. He reminded us that, as Edmund Gettier pointed out, knowledge is not the same as “justified true belief”, as philosophers had once thought. It seemed to me that perhaps a mathematical example of such a situation was the moment when Andrew Wiles, at the Newton Institute, announced Fermat’s Last Theorem. It is true, and he believed at that moment (with justification) that he had proved it. But subsequent events showed that he didn’t know that because the proof had a hole.

The final presentation was done remotely, via video and Skype, by Katie MacCallum from Brighton. This remarkable individual is an artist and a philosopher and is also knowledgeable about mathematics; she is working closely with nine mathematicians at various career stages, attempting to understand what they are doing whey they work, and to translate this understanding into works of art. The video combined her presentation with animated examples of knot diagrams drawn on a blackboard. Remarkably, a long segment of her talk consisted of a defence of the mathematicians’ customary use of blackboard and talk to communicate. She gave three reasons: briefly,

  • this method slows the presenter down to the audience’s speed, which often doesn’t happen with a computer presentation;
  • writing is quite hard work, and the presenter often pauses at the end of a line, and fills the silence with “meta-information” about the material, which aids understanding;
  • use of different parts of the board (the top, a box to one side) gives the audience clues to the structure of the material presented.

And so the meeting ended, since Ursula was not well enough to give the last talk. She encouraged us to walk on West Sands, but I slipped off home, to try to catch up on urgent tasks …

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Another Cameron–Erdős problem solved

In 1990, Paul Erdős and I published a paper on the topic of counting subsets of the first n natural numbers satisfying some restriction.

The case we worked hardest on, and which attracted the most interest (it has its own Wikipedia page) concerned sum-free sets, those containing no solution to the equation x+y = z. We conjectured that the number of such sets is asymptotically c.2n/2, where the constant c depends on the parity of n (that is, there are two different constants, one for even numbers and one for odd). We made substantial progress on this, and the proof was given independently by Ben Green and Sasha Sapozhenko. It was rather pleasing to me that we had divided the sets into three classes, and proved the result for two of the three; Ben Green showed that the third class is asymptotically smaller.

But we considered many other problems in the paper. Some we solved, but for many we were able to say something, though less than a solution.

One example was sets satisfying the condition that no element of the set divides any other. We conjectured that the number f(n) of such sequences grows exponentially, in the sense that (f(n))1/n tends to a limit as n→∞. By giving upper and lower bounds for f(n), we were able to show that the putative limit would be between 1.56 and 1.59. I thought this might be one of the first to fall …

I am happy to report that our conjecture has been proved by Rodrigo Angelo in the latest edition of Integers. It is a lovely proof, short but intricate; though my acquaintance with Paul was rather brief, I do believe that he would have enjoyed it.

Unfortunately the proof gives no information about the limit, either its numerical value or its status as rational, algebraic or transcendental. So still work to do.

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Integrals of groups

Everyone who has studied mathematics knows what the derivative and integral of a function are. The derivative measures rate of change, the integral (the inverse operation) measures area under a curve. They are inverse operations; and two functions have the same integral if and only if they differ by a constant.

In a group, the commutator is a function of two variables which is the identity if and only if the two arguments commute. The commutator subgroup of a group is the subgroup generated by the commutators; so it is the identity subgroup if and only if the group is abelian (all pairs of elements commute).

The commutator subgroup is also known as the derived group. I don’t know who introduced this terminology; it suggests an analogy between group theory and calculus which would never have occurred to me. (There is a closer analogy with differentiation in the relation between a Lie group and its Lie algebra, but that is a different story.)

On one of my visits to Lisbon, João Araújo and Francesco Matucci introduced me to what sounded like an interesting game: if forming the derived group is the analogue of differentiation in calculus, what can we say about integration? We will say that H is an integral of G if G is isomorphic to the derived subgroup of H.

The first thing that can be said is that a basic fact from calculus fails rather spectacularly here. The analogue of a constant function is an abelian group (one whose derived group is trivial); the analogue of adding a constant should presumably be taking the direct product with an abelian group. But, for example, among the integrals of the cyclic group of order 3 we have the dihedral group of order 6 and both non-abelian groups of order 27, which certainly do not differ by an abelian direct factor!

Nevertheless, we have been playing the game for a while, and have just posted a preprint on the arXiv.

The first thing we can say, which finite group theorists will appreciate, is that, if a finite group G has an integral, then it has a finite integral. For suppose that H is an integral of G. Choose pairs of elements of H whose commutators generate G; the subgroup of H generated by these elements is also an integral of G. So we may assume that H is finitely generated. Now conjugate elements of H lie in the same coset of the derived group; so the conjugacy classes in H have size bounded by |G|; that is, H is a BFC-group. In a finitely generated BFC-group, the centre has finite index, and so is finitely generated; so the centre is the direct product of a finite group and a torsion-free group. Factoring out the torsion-free group we obtain a finite group whose derived group is G.

Every abelian group is integrable. However, not every group is integrable. For example, for every even number n greater than 4, the dihedral group of order n has no integral. Of the two non-abelian groups of order p3, where p is prime, one is integrable and the other is not.

How do we decide whether a group is integrable? This leads to an interesting question to which we don’t have an answer:

Problem: Find a good upper bound for the order of the smallest integral of an integrable group of order n.

It is clear that there is such a bound. For there are only finitely many groups of order n up to isomorphism; we can take the largest order of the smallest integral of an integrable group in the collection. However, just knowing that a bound exists is no use for the obvious algorithm for testing integrability, viz., check all groups whose orders are multiples of n up to f(n), where f is the above function, to see whether any of them has derived group isomorphic to G.

The smallest integrals of the cyclic group of order 2 are the two non-abelian groups of order 8, and we conjecture that the function f satisfies f(n) ≤ n3 for all n. We can show that an abelian group of order n has integral of order at most n1+o(1), though finding an exact formula for the smallest integral is quite tricky; the right answer could conceivably be Cn for some constant C.

A result which may be folklore characterises the set of positive integers n such that every group of order n is abelian: they are precisely the integers n which are cube-free and for which there do not exist primes p and q such that either

  • p and q divide n, and q divides p−1; or
  • p2 and q divide n, and q divides p+1.

A proof by Robin Chapman can be found here or here.

Since every abelian group is integrable, one could ask for a similar characterisation of the set of numbers n for which every group of order n is integrable. This question is answered in our paper. The answer is almost the same as above; the second condition on the primes p and q simply has to be deleted. Thus, for example, there exist non-abelian groups of order 75, but every group of this order is integrable.

An interesting problem here is: does the density of the set of such numbers exist, and if so, what is it? Of the first hundred million positive integers, there are 32,261,534 for which every group of that order is integrable.

There are groups which can be integrated arbitrarily many times (that is, which are the nth derived group of some other group, for all n). For example, if we take the group of strictly upper triangular (2n+1)×(2n+1) matrices over a field F, the nth derived group is isomorphic to the additive group of F. However, we do not know an answer to the following question:

Problem: For which groups G does there exist a strictly ascending sequence G=G0<G1<… with each group the derived group of the following one?

If G is a perfect group (equal to its derived group), then of course such a sequence exists with the inequality replaced by equality or non-strict inequality (all terms equal to G); but with the strict inequality, we have no examples. Such a group must have non-trivial centre, since we show that for groups with trivial centre the sequence must terminate after finitely many steps.

All of this can be viewed in a different light, as an example of what might be termed “inverse group theory”. There are various constructions which start from a group and produce another group: very often the new group is a subgroup (for example, the derived group, Frattini subgroup, Fitting subgroup, centre), but it may be a quotient (soluble residual, derived quotient) or even something else (Schur multiplier or other cohomology group). I would like to say “functor”, but of course not all of these constructions are functorial in the sense of category theory.

For each of these constructions, there is an inverse problem: Given a group G, is there a group H such that the derived group (or Frattini subgroup, or soluble residual, or Schur multiplier) of H is G?

Some of these problems are trivial. For two examples,

  • The centre of a group is abelian. But every abelian group is the centre of a group (for example, itself).
  • The Fitting subgroup of a group is nilpotent. But every nilpotent group is the Fitting subgroup of a group (for example, itself).

We have by no means explored all the examples listed. But it appears that the question is non-trivial, and has interesting aspects, for derived group and for Frattini subgroup. Let me give one problem related to the latter.

True or false? If the (finite) p-group G is the Frattini subgroup of a group, then it is the Frattini subgroup of a finite p-group.

I mention in conclusion that not every p-group is the Frattini subgroup of a p-group.

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Portugal’s oldest university?

I am in Portugal, a country which is as near as can be my second home now. At the weekend, we went on an incredible trip to the Alentejo, the region of Portugal between the river Tejo and the Algarve. Normally a hot dry area, at the moment it is cool and green because of a lot of rain in the late winter. I won’t attempt to describe the whole trip but will focus on our first stop, the town of Évora, where I had two treats that I had missed on my previous visit to the town.

The first was the cathedral. Most famous for its treasurehouse of religious artefacts made of precious metal, works of art, richly embroidered robes, and so forth. More interesting to me was being able to walk around on the roof, and the cloisters with many Moorish patterns in the round windows.

But most startling was the University of Évora. As my title suggests, it is sometimes considered as a claimant for the title of Portugal’s oldest university, though the University’s own information leaflet only claims to be the second oldest. The other two claimants are Lisboa and Coimbra. It seems that Lisboa was founded first. But it disappeared at the very time that Coimbra was founded, and didn’t reappear until the twentieth century. (Or you could say, the University was moved from Lisboa to Coimbra. My Oxford college, Merton, was founded in the district of south-west London of the same name, and moved to Oxford ten years later: Simon de Montfort’s war was raging in 1264, but things were more peaceful ten years later. Which date is the College’s foundation date?)

Anyway, the university of Évora was founded by the Jesuits in 1553, becoming a University six years later as a result of a Papal bull, and closed in 1759 when the Marquis of Pombal expelled the Jesuits from Portugal. It was re-founded in 1973 as a public university. But their leaflet says on the front, “Universidade de Évora, 1559–2018”.

Inside the front gate is a large court, with a fountain in the middle and cloisters round the outside. Leading off the cloisters are doors into lecture halls. I will list the names since I find this quite remarkable.

  • The Geography Room.
  • The Metaphysical Philosophy Room.
  • The Physics Room.
  • The Greek Philosophy Room.
  • The Aeneid Room.
  • The Holy Scriptures Room.
  • The Geometry and Astronomy Room.
  • The Great Hall.
  • The Latin Room, “The Bucolics”.
  • The Literary Genres Room.
  • The Months of the Year (and Signs of the Zodiac) Room.
  • Hunting Scenes Room.
  • Scenes of the Countryside, Fishing and Hunting Room
  • And another of these.

I observe that there are only two subjects of the mediaeval Trivium and Quadrivium listed there. The absence of Music is particularly surprising since, in those days, Évora was renowned for its music, especially church music.

We looked in all the rooms as far as the Great Hall, which was used for candidates’ oral exams. Each room had azulejos (the celebrated Portuguese blue and white tiles) all the way round, with pictures relevant to the subject of the room. Thus, for example, the Physics Room had an illustration of the famous experiment by Otto von Guericke where he put two metal hemispheres together and pumped out the air; teams of horses were unable to pull them apart. (There is a sculpture illustrating the same experiment in the town of Magdeburg.)

But for me the most striking was the one shown below. This was in the metaphysical philosophy room. I am not sure whether it is a good slogan for metaphysical philosophy, but for mathematics it is near perfect.

Always Abstract

We went upstairs in search of the Library, which we eventually found. On the way we passed many professors’ offices of the modern University. Needless to say the subjects represented were rather different: history, social science, and literature.

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A counting problem

I quite often get questions by email. Mostly I can’t answer them, but sometimes I can, and sometimes the answers might be of more general interest.

Suppose that we have m boxes and a supply of identical balls. In how many distinct ways can we place n balls in the boxes? This is a standard combinatorial problem: we are choosing boxes from the set of m to place the balls in, and doing this n times, where the order of choice is unimportant and selections may be repeated. The answer is the binomial coefficient (m+n−1 choose n).

What if the boxes as well as the balls are indistinguishable? Then we can arrange them in a row so that the numbers of balls per box are non-increasing; the answer is the number of partitions of n into at most m parts; this is another standard problem, and while there is no simple formula for the solution, at least it is at least polynomial on residue classes.

The question I was asked was a common generalisation of these two. Suppose that a permutation group G acts on the set of boxes. How many ways of putting the balls into boxes, if two arrangements related by an element of G are not distinguished?

There is a standard piece of technology to answer such questions, the Cycle Index Theorem (in its simplest form). Take indeterminates s1,…,sm. For each element g of the permutation group, form the monomial in which the exponent of the power of si is the number of cycles of length i in the cycle decomposition of g. Then sum these monomials over all gG, and divide by |G|, the order of G; the resulting polynomial is the cycle index of G, denoted Z(G).

Now suppose that we have a set F of “figures”, each with a non-negative integer “weight”. The set of figures may be infinite, but we assume that there are only finitely many figures of any given weight. So the figures are described by a polynomial or formal power series A(x), in which the coefficient of xn is the number of figures of weight n.

We are interested in decorating the elements of the set X on which G acts with figures, and counting such “configurations” up to the action of G. Such a configuration is a function from X to F; the group G acts on the set of functions in a natural way, and we want to count the number of orbits of G on functions whose total weight (the sum of the weights of its values is given. So there is another polynomial or formal power series B(x), in which the coefficient of xn is the number of G-orbits on functions of weight n.

Now the Cycle Index Theorem states that B(x) is obtained from the cycle index Z(G) by replacing si by A(xi) for i = 1,…m.

In our problem, we take a figure of weight i, for each non-negative integer i, to mean that there are i balls in the box to which that figure is attached. So a function is an allocation of balls to boxes, and its weight is the number of balls attached. The figure-counting series is given by A(x) = 1+x+x2+… = (1−x)−1. Applying the Cycle Index Theorem gives the configuration counting series B(x).

For example, let G be the trivial group, with cycle index s1m. Then the generating function for the original balls-in-boxes problem is (1−x)m. Applying the Negative Binomial Theorem gives the result with which I began.

Extensions are possible. For example, if there are r distinguishable colours of balls, then the figure-counting series is (1−x)r, or (if we wish to keep track of the number of balls of each colour) the product of (1−xi)−1 for i = 1,…,r.

There is another interpretation, close to my heart, which I will treat in somewhat less detail. This involves oligomorphic permutation groups, those for which the number of orbits on n-element subsets of the domain is finite for all n. The counting problem here is exactly the same as counting orbits on n-sets of the group S Wr G, the wreath product of an infinite symmetric group S with G. The domain is partitioned into m infinite sets, and we have symmetric groups acting independently on each of these; elements of G permute these blocks among themselves. Now an n-element set of the domain is specified, up to the action of the symmetric groups, by giving the cardinality of its intersection with each part; and we want to count the orbits of G on such m-tuples summing to n. So the orbit counting problem is identical with the original balls-in-boxes problem. Now a standard formula for counting orbits of a wreath product gives the answer, which turns out to be the same as the one obtained using the Cycle Index Theorem.

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