charts - Best visualization for combinations

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 2 and 2 genes from parents (with 1 shared), which makes for 2^2 = 16 possibilities.

The current UI shows the list of possibilities, but nothing visually conveys the magnitude. 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 maybe(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).

Thanks!