Unimodality of regular partition polynomials

Abstract

Let n, p and j be integers. Define

$$\begin{aligned} R_{n,p,j}(q):=\prod _{k=0}^{n}(1+q^{pk+1})(1+q^{pk+2})\cdots (1+q^{pk+j}). \end{aligned}$$

The coefficients of the polynomial \(R_{n,p,j}(q)\) count certain regular partition. Recently, Dong and Ji studied unimodality of the polynomials \(R_{n,p,p-1}(q)\). As an extension, in this paper, we give a criterion for unimodality of the polynomials \( R_{n,p,j}(q)\) for \(p \ge 6\) and \(\lceil \frac{p+1}{2}\rceil \le j\le p-1.\) In particular, using our criterion and Mathematica, we obtain that \(R_{n,p,j}(q)\) is unimodal for \(n\ge 3\) if \(6\le p \le 15\) and \(\lceil \frac{p+1}{2}\rceil \le j\le p-1.\)