Questions tagged [combinatorial-proofs]

I want to know if there exist a known expansion or can be derived of the polynomial $$ \prod_{i=1}^{m}\prod_{j= 1}^{n}(1-z(x_i + y_i))^{-w} \tag{*}$$ in terms of Schur function. That is asking for (*) ...

Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means: $$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j} =\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$ QUESTION. Can you provide a ...

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

Let a and b be coprime positive integers. Then the number a+b divides the binomial coefficient ${a+b \choose a}$. I know how to prove this combinatorially - for example after choosing an ordered set ...