Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than 1, that is, they are not relatively prime. There are 3718 partitions of 28 of which 291 are partitions into five parts. I have been told that there is just one of these that can be uniquely recovered from its P-graph. Which is it?
2026-02-22 17:15:19.1771780519
A certain partition of 28
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Using a computer search, of the $291$ partitions of $28$ into $5$ parts, the only one whose associated $P$-graph has degree sequence is $3,3,2,1,1$ is $$3 + 4 + 5 + 6 + 10 = 28$$ hence, assuming you know that only one of the partitions is recoverable from its $P$-graph, then that must be the one.