Stanley's conjectures on the Stern poset

Abstract

The Stern poset S is a graded infinite poset naturally associated with Stern's triangle, which was defined by Stanley, in analogy with Pascal's triangle. Stanley noted that every interval in S is a distributive lattice. Let Pn denote the unique (up to isomorphism) poset for which the set of its order ideals, ordered by inclusion, is isomorphic to the interval from the unique element of row 0 of Stern's triangle to the n-th element of row r for sufficiently large r. For n≥1 letLn(q)=2⋅(∑k=12n−1APk(q))+AP2n(q), where AP(q) represents the corresponding P-Eulerian polynomial. For all n≥1 Stanley conjectured that Ln(q) has only real zeros and L4n+1(q) is divisible by L2n(q). In this paper we obtain a simple recurrence relation satisfied by Ln(q) and affirmatively solve Stanley's conjectures.