  We say a polytope is Ehrhart positive if all the coefficients in its Ehrhart
polynomial are positive. Answering an Ehrhart positivity question posed on
Mathoverflow, Stanley provided an example of a non-Ehrhart-positive order
polytope of dimension $21$. Stanley's example comes from a certain family of
order polytopes. In this paper, we study the Ehrhart positivity question on
this family of polytopes. By giving explicit formulas for the coefficients of
the Ehrhart polynomials of these polytopes in terms of Bernolli numbers, we
determine the sign of each Ehrhart coefficient of each polytope in the family.
  As a consequence of our result, we conclude that for any positive integer $d
\ge 21,$ there exists an order polytope of dimension $d$ that is not Ehrhart
positive, and for any positive integer $\ell$, there exists an order polytope
whose Ehrhart polynomial has precisely $\ell$ negative coefficients, which
answers a question posed by Hibi. We finish this article by discussing the
existence of lower-dimensional order polytopes whose Ehrhart polynomials have a
negative coefficient.
