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

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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.

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  • $\begingroup$ Thank you so much! $\endgroup$
    – Mihawk
    Apr 28, 2020 at 6:47

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