A 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

A 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.