Unimodality and the reflection principle (Q2715939)

A sequence of numbers \((a_k)_{k\geq 0}\) is unimodal if \(a_0\leq a_1\leq \cdots \leq a_{m-1}\geq a_m\geq a_{m+1}\geq \cdots \) for some \(m\). A method for proving unimodality is presented: Interpret \(a_k\) as a number of certain lattice paths and use the reflection principle to get an injection (i.e. the \(\leq \) relation for cardinalities). The method is demonstrated on binomial coefficients, their products, and other examples. A result connected to a conjecture of R. Simion is given.