If I come up with a situation that works for one side of a combinatorial proof, does some interpretation always exist for how the other side counts that same situation? Or is it possible that one side cannot count a situation that the other side can count, even though they are numerically equivalent?
2026-03-26 01:34:33.1774488873
Can any counting situation that works for one side of an identity work for the other (combinatorial proof)
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Take for example the following identity. $$\frac{(2n)!}{n!(n+1)!}=4\binom{2n-1}{n}-\binom{2n+1}{n}.$$ The left side has a smooth interpretation.
How to understand this interpretation from the right side?