On the zeros of some combinatorial polynomials

Let $a_n(q)$ denote the distribution on the set of involutions of size $n$ for the statistic which records the number of fixed points. We show for a range of $q$ values that the polynomial $\sum_{i=0}^n a_i(q)x^i$ always has the smallest possible number of real zeros, that is, none when the degree is even and one when the degree is odd. On the way, we show that the sequence $a_n(q)$ is log-convex for all $q\geq1$. Our proof in the case $q=1$ is elementary, while the proof for the general case makes use of programming to estimate the zeros of some related analytic functions in $q$. Furthermore, the sequence of real zeros obtained from the odd case is shown to be monotonically convergent. We also consider the polynomial $\sum_{i=0}^n d_ix^i$, where $d_i$ denotes the number of derangements of an $i$-element set, and show that the same holds for it. Our results extend recent ones concerning Fibonacci and Tribonacci coefficient polynomials.