If $a_n$ satisfies the necklace congruence, you can verify that $\exp \left( \sum_{k \ge 1} a_k \frac{t^k}{k} \right) = \prod_{n \ge 1} (1-x^n)^{-\frac{b_n}{n}}$, where $b_n$ is an integer sequence given by $b := a * \mu$, and so $b(n) \equiv 0 \mod n$ and the product must have integer coefficients.

Other direction: Assume $\exp \left( \sum_{k \ge 1} a_k \frac{t^k}{k} \right)$ has integral coefficients. Again, one can write it as $\prod_{n \ge 1} (1-x^n)^{-\frac{b_n}{n}}$ where $b = a * \mu$. Because the coefficient of $x^n$ is $\frac{b_n}{n}+$ a sum consisting of binomial coefficients depending on $\{ \frac{b_i}{i} \}_{i<n}$, an induction arguments shows that $\frac{b_n}{n}$ must be integral for all $n$, and so $a_n$ satisfies the necklace congruence.

]]>This sounds close to what I want. Let me explain the symmetric function idea, although it doesn’t work: ideally the sum on the RHS should result from an application of Burnside’s lemma, but this doesn’t actually work out since the term corresponding to the identity is smaller than the other terms in general. Even if that’s not the case, there should be some way to explain why dividing by on the RHS gives an integer.

As for the other idea, for a permutation representation is an average over a bunch of terms coming from adjacency matrices of functional graphs describing the action of on some finite set. Molien’s theorem relates these to homogeneous invariants, so at least for this case one should be able to give a basis for these invariants which is in bijection with whatever it is counts.

]]>Here is an argument. The trace of A^n is (as you point out) the number of loops in A^n with a specified start-and-end vertex. The trace of A^n/n feels like the number of loops in G with no such specified vertex, but A^n/n can fail to be an integer. A more careful way to put it is: A^n * (n-1)! is the number of structures of the form

– cyclic ordering of {1,…,n}

– map from {1…n} to G preserving the cyclic ordering.

The exponential generating function of this is the logarithm of 1/det(1 – t A) (the second displayed formula in your “Zeta functions” section). A standard argument about exp-of-an-exponential-generating-function says that the coefficients of t^n/n! in 1/det(1 – t A) itself count structures of the form

a- decomposition of {1,…,n} into pieces, and a cyclic ordering of each piece

b- map from each cyclically-ordered piece into G preserving the order

Or “maps from a smooth graph with n vertices into G” for short. (a- can be recast as “total ordering of {1…n}” using the cycle notation for permutations)

Can you expand your comment about the symmetric function identity? Molien’s theorem?

]]>