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Combinatorial proofs include, for example, proofs using double counting or bijective proofs.
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Proofs by Schubert calculus and combinatorics
Unimodality of partitions inside a box of size $m\times n$ (i.e. partitions with number of parts at most $m$ and size of the largest part at most $n$) follows from applying Hard Lefschetz Theorem to t …