Polynomials with palindromic and unimodal coefficients

Let $f(q)=a_rq^r+\cdots+a_sq^s$, with $a_r\neq 0$ and $a_s\neq 0$, be a real polynomial. It is a palindromic polynomial of darga $n$ if $r+s=n$ and $a_{r+i}=a_{s-i}$ for all $i$. Polynomials of darga $n$ form a linear subspace $\mathcal{P}_n(q)$ of $\mathbb{R}(q)_{n+1}$ of dimension $\lfloor{n/2}\rfloor+1$. We give transition matrices between two bases $\left\{q^j(1+q+\cdots+q^{n-2j})\right\}, \left\{q^j(1+q)^{n-2j}\right\}$ and the standard basis $\left\{q^j(1+q^{n-2j})\right\}$ of $\mathcal{P}_n(q)$. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.

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