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> Combinatorics gives us lots of ways to calculate the sizes of sets, and in my experience, they rarely if ever involve an explicit enumeration. I don't think
by JadeNB 5y ago
> Combinatorics gives us lots of ways to calculate the sizes of sets, and in my experience, they rarely if ever involve an explicit enumeration.
I don't think that this is true. I think that explicit enumerations are rarer than exact size computations, which are themselves rarer than asymptotic size estimations, for presumably obvious reasons; but a bijective proof (https://en.wikipedia.org/wiki/Bijective_proof https://en.wikipedia.org/wiki/Bijective_proof) is still regarded as the gold standard in combinatorics, and as a desideratum.
- civilized 5y agoA bijective proof isn't the same as an enumeration, which is a bijection to the set 1,2,3... Of course you can derive an enumeration by bijection to a set with an existing enumeration, but the point is that that's usually not of interest.