Question

I'm not entirely sure I get the question, but I think the theory of partitions has many examples of the kind of thing you want. The number of partitions of $n$ into (say) 17 parts equals the number of partitions of $n$ into parts the largest of which is 17. There's a bijection between the two sets of partitions; the bijection is so natural (once you've seen it) that it's tempting to say we're just talking about one set of partitions but looking at it two different ways. The number of partitions (of $n$) into odd parts equals the number of partitions into distinct parts; here, too, there's a bijective proof, but it's quite a bit harder to find. There is a host of these things in the Andrews and Eriksson book, Integer Partitions (and in many other places where partitions are discussed in detail).