Earlier I mentioned that the theory of Young tableaux is the source of one of my favorite proofs. Today I’d like to present, again from the theory of Young tableaux, one of my favorite **pairs** of proofs.

A standard Young tableau is a chain in Young’s lattice; equivalently, it is a sequence of Young diagrams, each of which has exactly one more box than the previous. One can succinctly write down such a chain by taking the last Young diagram in the chain and recording the “path” traced out by the rest of the chain: write down a 1 in the box corresponding to the first Young diagram, a 2 in the box added by the next Young diagram, and so forth. In other words, a standard Young tableau is a Young tableau filled with positive integers which is strictly increasing across rows and columns.

Given a Young diagram , let denote the number of standard Young tableaux of shape ; equivalently, one can define to be the poset of Young diagrams fitting inside , and then is equal to the number of maximal chains of .

Finally, one last bit of notation. If is a Young diagram for a partition of , we will write either or . Then we have the following beautiful result.

**Theorem:** .