I would like to create statistical test that detects over-representation / enrichment of structures such as pairs, triplets, quadruplets, etc. in a given group of obejcts compared to another group.

The tricky part is to take into account the substructures. For example, let's say I'm interested in overrepresentation of triplets or quadruples of object and I suppose I should somehow consider/enclose information about lower structures like pairs or even the frequencies of individual objects.

But let's start at the very beginning. Let's say that we have N groups represented by an ordinary arrays of length $L_i$.
We also have colourful balls arranged in a certain way in groups cells, balls may co-occupy the same cell.
The whole situation is given and we have full information about it.

(If it would facilitate discussion, we can say that we know probabilities of occurence for individual objects / balls for each cell.The same ball has different probabilities in different cells and balls of different colour have different probabilities in the same cell.
But for simplicity I don't use this here.)

Now, the example for pairs. I'm particularly interested in pair of blue and red balls that preseve a fixed structure composed of: an order: blue is on the left hand side and red on the right and a distance: distance between this two cells containing this balls is set to e.g. 4.

Subsequently, I choose one group, let's say $G_1$ wich will be called foreground and the remaining groups together will form the background.
I'm looking for the statistical test for checking

null hypothesis $H_0$: the probability of occurrence of structured (maintained order and distance) pair of blue and red balls is the same for
the foreground group $G_1$ and the background (othe other groups all together).

against

alternative hypothesis $H_A$: probability of such structured pair is higher in the foregorund.

In the most trival approach, I can say that occurrence of such structured pair anywhere in the foreground is equally likely, with the same
unknown probability $p_f$ for each cell. The same applies to background and unknown probability $p_b$.
After that, we get sequences of independent random variables $F_i \sim Bernoulli(p_f)$ and $B_i \sim Bernoulli(p_b)$.
$F_i = 1$ denotes occurrence of structured pair starting at $i$-th cell in the foreground group.
All groups are long, so from Central Limit Theorem we get that the mean of $F_i$ and the mean of $B_i$ have Normal distribution.
$$\text{(mean of $F_{i}$)} ~~~~~~ \frac{\sum\limits_{i} F_i}{L_1} \sim Normal(p_f, \frac{p_f \cdot (1 - p_f)}{L_1} )$$
$$\text{(mean of $B_{i}$)} ~~~~~~ \frac{\sum\limits_{i} B_i}{\sum\limits_{2 \leq i \leq N} L_i} \sim
Normal(p_f, \frac{p_b \cdot (1 - p_b)}{\sum\limits_{2 \leq i \leq N}} L_i)$$

Because I don't know anything about variations, thus I use Welch's T test.
But in my question the selection Welch's T test is not crucial.
Moreover, I don't know even whetehr this method is correct, but that's not the problem.
The main problem lies here:

In the test above, there's nothing special about pairs. I would like to adapt this (or whichever other test) to handle triplets. I feel really bad to literally replace pairs with triplets.

I have inetrnal premonition, that I should take into account lower structures like pairs and even the frequencies of individual objects of which this triplets are composed.

I will be very grateful for any hints and sugesstions how to cope with over-representation / enrichment of bigger structures.