I have an interesting puzzle that I am unable to solve.
Preliminaries: Suppose that $\mathbb R^2$ is given a colouring $\mathbb R^2 \to \{0, \ldots, k-1\}$. For a given distance $d$, we can compute the transition probability by considering arbitrarily many line segments of length $d$, recording the colour at both ends and taking the average. In this manner we obtain a transition matrix $T_d$ in which the entries $T_{d,(i,j)}$ correspond to the probability of encountering colour $j$ at the end of a line segment that starts at a point with colour $i$. From this, one can compute the conditional probability $\mathbb P_i(j)$ of encountering colour $j$ at distance $d$ away from $i$ (sorry for the abuse of notation).
The problem: Given a finite metric space $X$ consisting of $n$ points, a colouring $C : X \to \{0, \ldots, k-1\}$ and transition matrices $T_d$ for all pairwise distances $d$ in $X$, what is the probability of encountering colouring $C$ when we embed $X$ isometrically into $\mathbb R^2$?
What I have tried: To keep things simple, I considered 3 equidistant points $X = \{ x_1, x_2, x_3\}$ (so we need only a single transition matrix $T$) and 3 possible colours. Suppose that our colouring is $C(x_1) = c_1, C(x_2) = c_2$ and $C(x_3) = c_3$. I figured that the probability of this colouring occurring is invariant under isometries (in this case given by the dihedral group D3) and hence I tried (amongst many other things, most of them wild guesses rather than reasoned approaches):
$$P(C | X) = T(c_1, c_2) (\mathbb P_{c_2}(c_3)\mathbb P_{c_3}(c_1) + \mathbb P_{c_1}(c_3) \mathbb P_{c_3}(c_2)),$$
and
$$P(C | X) = \mathbb P_{c_1}(c_2) \mathbb P_{c_2}(c_3) \mathbb P_{c_3}(c_1) + \mathbb P_{c_1}(c_3) \mathbb P_{c_3}(c_2) \mathbb P_{c_2}(c_1).$$
But both are incorrect: summing these probabilities over all possible permutations does not add up to 1. Ideally, (for computational reasons), I would like to be able to compute the probability for an isometry class without having to compute the number of distinct colourings within a class. I am unsure how to proceed and I am grateful for any help. Thanks in advance!