论文标题
斯坦利的猜想在船尾poset上
Stanley's conjectures on the Stern poset
论文作者
论文摘要
船尾poset $ \ Mathcal {s} $是与Stern的三角形自然相关的无限姿势，斯坦利（Stanley）与帕斯卡尔（Pascal）的三角形相似。令$ p_n $从行的独特元素$ 0 $ 0 $ 0 $ r $ $ r $的$ n $ th $ n $ th $ n $ th $ n $ r $表示$ \ mathcal {s} $的间隔。对于$ n \ geq 1 $ let \ begen {align*} l_n（q）＆= 2 \ cdot \ left（\ sum_ {k = 1}^{2^n-1} a_ {p_k} a_ {p_k}（q）（q）\ right）相应的$ p $ -eulerian多项式。对于任何$ n \ geq 1 $ $ stanley猜想，$ l_n（q）$只有真实的零，$ l_ {4n+1}（q）$可以除以$ l_ {2n}（q）$。在本文中，我们获得了一个简单的复发关系，可以通过$ l_n（q）$满足，并肯定地解决了斯坦利的猜想。我们还建立了$ l_n（q）$的系数的渐近正态性。
The Stern poset $\mathcal{S}$ is a graded infinite poset naturally associated to Stern's triangle, which was defined by Stanley analogously to Pascal's triangle. Let $P_n$ denote the interval of $\mathcal{S}$ from the unique element of row $0$ of Stern's triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $n\geq 1$ let \begin{align*} L_n(q)&=2\cdot\left(\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 $n\geq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this paper we obtain a simple recurrence relation satisfied by $L_n(q)$ and affirmatively solve Stanley's conjectures. We also establish the asymptotic normality of the coefficients of $L_n(q)$.
