2020/05/31 by Arthur L. B. Yang, Yang, Arthur L. B.
Mathematics · #05A15 #11B37 #26C10 #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2006.00400
openalex publication_date 2020/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Stern poset S is a graded infinite poset naturally associated to Stern's triangle, which was defined by Stanley analogously to Pascal's triangle. Let Pn denote the interval of S 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 let Ln(q)amp;=2⋅(∑k=12n-1APk(q))+A_P2n(q), where AP(q) represents the corresponding P-Eulerian polynomial. For any 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. We also establish the asymptotic normality of the coefficients of Ln(q).