On the factorization of Eulerian polynomials (Q790870)

The Eulerian polynomials \(P_ k(x)\) can be defined by setting \(R_ k(x)=\sum^{k}_{s=0}A_{ks}x^ s\) where \(A_{ks}=\sum^{s}_{j=0}(-1)^ j\left( \begin{matrix} k+1\\ j\end{matrix} \right)(s-j)^ k,\) and then \(P_ k(x)=R_ k(x)\) if k is odd, \(P_ k(x)=R_ k(x)/(x+1)\) if k is even. It has been conjectured that \(P_ k(x)\) is irreducible over the rationals. The author obtains partial results in this direction by showing that \(P_ k(x)\) must have an irreducible factor of degree at least 11 k/14 and improving this lower bound, even proving the conjecture, for certain special values of k.