Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Is there a reference for the following lemma (which is useful in counting unlabeled k-trees)? It seems to me that it should be known, but I haven't been able to find it anywhere.

Let $G$ be a finite group acting on a weighted set $S$. We assume that each weight is a product of powers of variables. We do not assume that $S$ is finite, but we do assume that the sum of the weights of the elements of $S$ converges as a formal power series.

Let $M(S)$ be the set of multisets of elements of $S$. We may extend the action of $G$ on $S$ in an obvious way to an action on $M(S)$. We define the weight of a multiset in $M(S)$ to be the product of the weights of its elements. For any $g\in G$, we denote by $\operatorname{fix}(g)$ the sum of the weights of the elements of $S$ fixed by $G$. For any formal power series $u$ in the variables that appear in the weights, we define $p_n[u]$ to be the result of replacing each variable in $u$ by its $n$th power.

Let $g$ be an element of $G$. Then the sum of the weights of the elements of $M(S)$ fixed by $g$ is \begin{equation} \exp\biggl(\sum_{m=1}^\infty \frac{p_m[\operatorname{fix}(g^{m})]}{m} \biggr). \end{equation}

This lemma is easy to prove by reducing to the case in which $g$ acts transitively on $S$, so I'm not looking for a proof, just a reference. (The case in which $g$ is the identity is well known.)

share|improve this question

1 Answer 1

Thanks to the OEIS, I found a reference to almost the same formula:

Marko V. Jarić and Joseph L. Birman, New algorithms for the Molien function, Journal of Mathematical Physics 18, 1456 (1977); doi: 10.1063/1.523442; http://dx.doi.org/10.1063/1.523442

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.