WTF is a Number Bond?

Not chemistry, as it happens. I was searching for similar images to one of my line drawings (always fun) and came across these 'number bond things' :

The one on the left - hilariously - is just "1 + 1 = 2". Ok, so that's a deliberately jokey example; real ones have larger numbers and one of the three numbers is for the student to fill in. On the right is a more complex example, drawn as a DAG (directed acyclic graph) although at least one of the example I saw had a node at the bottom with three parents!

In any case, what these things really are representing is partitions of numbers - which are usually drawn as Ferrer's diagrams (or Young tableaux) which I'll refer to as "Ferrer's-Young diagrams". These have a superior feature as shown here:

So one FY-diagram can represent two different number bonds. Note that I've made the crazy leap of making number bonds with more than two parts (or 'addends'). Clearly 1+1+3+4 = 9 = 1+2+2+4 as these partitions are a transposed pair.

One more thing occurs to me - what happens if you do this on a graph, not just a tree? You can label the leaves with - say - an increasing sequence of numbers, and then move to the parents, summing as you go. To work, this algorithm has to do something like add in the largest number from the previous round to get unique numbers:

Here we are labelling the leaves with the numbers {1, 2, 3} and then their parents with {1+3, 2+3+3} and then the final one with {5+8+8}. Of course this labelling is not canonical - you would have to try all permutations of leaf labels. However it's quite nice to see a connection between something as simple as number bonds and something more complex like vertex labelling!

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