Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are nonnegative and such that $p(z)$ is rational (and hence a rational integer) whenever $z$ is an $n$th root of 1?
Two examples should clarify the question: $p(z) = \Phi_5(z)$ $=$ $1 + z + z^2 + z^3 + z^4$ is rational whenever $z$ is a 4th or 5th root of 1, while $p(z) = \Phi_5(z) \Phi_6(z)$ $=$ $1 + z^2 + z^3 + z^4 + z^6$ is rational whenever $z$ is a 5th or 6th root of 1.
I imagine that there are general conditions (some trivial and some less so) under which rationality is guaranteed, as well as sporadic cases under which rationality holds "by accident".
Ideally I would like not just a criterion but a generic construction (applying whenever the criterion holds) of a permutation $\pi$ of some set of cardinality of $p(1)$ such that for all integers $k$ the number of fixed points of $\pi^k$ equals the magnitude of $p(\zeta^k)$ where $\zeta$ is a primitive $n$th root of 1; or, failing that, an invertible linear map $A$ from some $p(1)$-dimensional vector space to itself such that for all integers $k$ the trace of $A^k$ equals $p(\zeta^k)$. (E.g., for the four pairs $p(\cdot),n$ mentioned above, the permutations $\pi$ should have respective cycle-types (1)(2345), (12345), (12345), and (12)(345).)
Such a construction would be relevant to the Galois-theoretic perspective on the cyclic sieving phenomenon. (The CSP was introduced by Reiner, Stanton, and White in 2004 and is discussed for instance in http://arxiv.org/abs/1008.0790 and http://www.ams.org/notices/201402/rnoti-p169.pdf .)
