In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all $\ell,m\geq 8$ and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

翻译：2013年,Pak和Panova证明,基于Youngcoaux的组合体和Kronecker系数的半组特性,美元-二元系数(以美元计)的严格单元属性为$binom = ell+m ⁇ m ⁇ qqq$(以美元计),它们表明,所有美元、m\geq 8美元和其他一些情况都是真实的。基于计算机代数,我们提出了解决这一问题的不同方法,我们为这些多元值的系数建立封闭形式,然后使用圆柱形代数分解法,以精确确定严格单式的系数范围。这一战略使我们能够应对这一问题的概括性,例如,用较大差距或相关序列的单一性来显示问题。特别是,我们提出了斯坦利和扎内洛另外两个同位词的例子的证据。