# Counting connected fundamental domains of actions on Cayley graphs

The following question arises, for me, from mathematical music theory:

Write $({\Bbb Z}^n,E_n)$ for the Cayley graph of ${\Bbb Z}^n$ relative to standard free generators.

Given a subgroup $L$ of ${\Bbb Z}^n$ of finite index, how ought one efficiently count and/or enumerate, up to translation, the connected (as vertex-induced subgraphs) fundamental domains for the action of $L$ on $({\Bbb Z}^n,E_n)$.

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An inefficient way for enumerating all connected fundamental domains is by remarking that they are all contained (up to translation) in the ball of radius at most $(l+1)/2$ (with respect to word length in generators) at the origin where $l$ is the index of the subgroup $L$. It is thus enough to consider all connected subgraphs having $l$ vertices contained in this ball and to check that vertices of such a subgraph represent distinct classes modulo $L$.
A perhaps more effective idea is as follows: Call two connected fundamental domains $D_1$ and $D_2$ adjacent if $D_1\cap(D_2+A)$ contains $l-1$ elements for a suitable vector $A$ (where $l$ is the index of the subgroup). This turns the set of all connected fundamental domains (considered up to translation) into a finite graph.