Stanleys conjectures on the Stern poset

The Stern poset $mathcal{S}$ is a graded infinite poset naturally associated to Sterns triangle, which was defined by Stanley analogously to Pascals triangle. Let $P_n$ denote the interval of $mathcal{S}$ from the unique element of row $0$ of Sterns triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $ngeq 1$ let begin{align*} L_n(q)&=2cdotleft(sum_{k=1}^{2^n-1}A_{P_k}(q)right)+A_{P_{2^n}}(q), end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $ngeq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this paper we obtain a simple recurrence relation satisfied by $L_n(q)$ and affirmatively solve Stanleys conjectures. We also establish the asymptotic normality of the coefficients of $L_n(q)$.