论文标题
纳拉亚纳数字改进的对称性的组合证明
A combinatorial proof of a symmetry for a refinement of the Narayana numbers
论文作者
论文摘要
我们建立了一定数字的诱人对称性，以完善Narayana数字。就戴克路径而言，这种对称性的解释是通过以下方式解释的：如果$ w_ {n，k，m} $是$ k $ cob $ ud $和$ m $ coptress的dyck路径的数量，$ ud $ ud $ ud $ ud $ ud $，则$ w_ w_ w_ {2k+1，k，k，m}我们给出了这一事实的组合证明，依赖于周期引理，并表明$ w_ {2k+1，k，m} $的数字是Narayana数字的倍数。我们证明了一个更一般的事实，以建立数字$ w_ {n，k，m} $与由于卡兰（Callan）导致的广义纳拉亚纳（Narayana）数字的关系。更通用数字的封闭形式的表达$ w_ {n，k_ {1}，k_ {2}，\ ldots，k_ {r}} $计数semilength- $ n $ n $ dyck路径$ k_ {1} $ k_ {1} $ $ $ $ ud $ -factors，$ k_ {2}还获得了$ k_ {r} $ $ u^{r} d $ - factors，以及在所有上升运行的情况下，对这些数字讨论过的对称性的更通用形式。最后，我们研究了多项式的属性$ w_ {n，k}（t）= \ sum_ {m = 0}^k w_ {n，k，m} t^m $，包括真实性，$γ$ pAsitivity，positivity--启动率和对称解体。
We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$, then $w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $w_{2k+1,k,m}$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $w_{n,k,m}$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $w_{n,k_{1},k_{2},\ldots , k_{r}}$ counting the semilength-$n$ Dyck paths with $k_{1}$ $UD$-factors, $k_{2}$ $UUD$-factors, $\ldots$ , and $k_{r}$ $U^{r}D$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$, including real-rootedness, $γ$-positivity, and a symmetric decomposition.
