How to visualize combinations of data for a gene discovery application?

What kind of a visualization/chart should I use for showing all the ways you can choose from a set of items? (i.e., number of possible combinations)

Concretely, I am showing potential offspring from two animals, where each parent may possess some number of genes, and the offspring inherits 0, 1, or both parent genes of each type. The genes have fun names (e.g., fire), and sometimes combinations of those genes have their own names (fire + pastel = firefly), but this is beside the point.

Here's a simple example that shows 3 and 2 genes from parents (with 1 shared), which makes for 2^5 = 32 possibilities.

The current UI (below) shows the list of possibilities, but nothing visually conveys the magnitude (In this case each is equally likely 1/16, but in many cases some possibilities will be 2/16 or 4/16). In other words, nothing about the "graphic" visually shows the user how likely each outcome is. Secondly, it would be great if the outcomes which share commonality (i.e., contain same genes) could be visually related.

My idea is something like a diamond shaped graph, or layered network, where at the top is the outcome where all genes are chosen, and below that a row of nodes with N-1, and so forth until the bottom row has 0 selected. Edges would connect the nodes beween layers with shared genes. Size of nodes could indicate probability. Something like this graph (but ignore the data).

I'm aware of Punnett Squares, but I'm not sure it's the best for combinations of this order (for one it doesn't not combine equivalent outcomes).

Update

Number of results depends on how many genes the parents have combined (2^N). I expect most of the time it will be with 1-4 gene parents each, so not more than 7 genes total, or 2^7 = 128 possibilities. Also, if there is any duplication between parents this is less. For example with 2 of the 7 being shared genes, that makes 54 possibilities. (See this example live). So most of the time I think 10-50 results.