# Proofs by Schubert calculus and combinatorics

Do you know some examples proved by two different methods: 1. Schubert calculus, 2. combinatorial method.

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 the Grassmann variety $$G_{m+n,n}$$ (see the paper 'Combinatorial Applications of the Hard Lefschetz Theorem' by Stanley). A combinatorial proof of this was given by Kathy O'Hara (reference: Unimodality of Gaussian coefficients: a constructive proof, J. of Comb. Theory, Ser. A, 53 (1990) 29-52.