For \(k=1,2,\dots\) denote by \(S_k\) the set of all polynomials \(f\) with rational coefficients having the property that \(f\) and its derivatives \(f',f'',\dots,f^{(k)}\) are integer-valued at rational integers. A characterization of polynomials in \(S_1\) has been given by \textit{L. Carlitz} [Indag. Math. 21, 294-299 (1959; Zbl 0100.27102)] and the authors prove an analogue of Carlitz's result for arbitrary \(k\). They also describe the pairs \(f,g\in S_k\) for which the ratio \(f/g\) is a polynomial belonging to \(S_k\). This generalizes a result of \textit{D. A. Lind} [Am. Math. Mon. 78, 179-180 (1971)].