# 学术报告二十六：杨立波— Stanley's two conjectures on Stern's triangle

Stern's triangle was introduced by Stanley, which is analogous to Pascal's triangle and naturally encodes a poset structure, called Stern's poset. Let $\{b_n(q)\}_{n\geq 1}$ be a sequence of polynomials  which appear as the Eulerian polynomials associated to Stern's poset. Stanley further studied the following polynomials

L_n(q)=2 (\sum_{k=1}^{2^n-1}b_k(q) + b_{{2^n}}(q),  n\geq 1.

Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this talk, I will show how to prove these two conjectures.