含缺陷格路的加细
Refinements of Lattice paths with flaws
摘要：
经典的Chung-Feller定理[2]告诉我们，长度为$n$且带有$m$缺陷的Dyck路的个数是$n$阶Catalan数，且与$m$无关.本文研究了带缺陷的Dyck路的四个参数加细问题，即峰、谷、双下降和双上升。令${p}_{n,m,k}$为具有$k$缺陷和$m$峰的半长度$n$的所有Dyck路径的数目。首先，我们推导出多项式$P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$的互易定理。然后我们得到了$p_{n,m,k}$与$p_{n,m,n-k}$之和的Chung-Feller性质。最后，我们给出了长度为$n$且具有$k$双上升的Dyck路的Chung-Feller型定理：具有$m$缺陷和$k$双上升的半长度为$n$的Dyck路的个数等于具有半长度为$n$，$k$双上升且不通过x轴以下的Dyck路的个数，这个个数由Narayana数计算.令${v}_{n,m,k}$（分别$d_{n,m,k}$）是具有$m$缺陷和$k$谷（分别为$m$和$k$谷）的半长度$n$的所有Dyck路径的数量。双下降）。得到了一些类似的结果。
英文摘要：
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ valleys (resp. double descents). Some similar results are derived.
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