We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply this general result to the enumeration of certain subsets of Dyck, Motzkin, skew Dyck, and skew Motzkin paths, defined recursively according to the first return decomposition with a monotonically non-increasing condition relative to the maximal ordinate reached by an occurrence of a given pattern $\pi$.
翻译：我们认为,$A_k(x)=p(x)A ⁇ k-1}(x)(q)(x) ⁇ sum ⁇ i=0 ⁇ k A_i(x)美元)的等式体系是美元=1美元,如果A_i(x)$0\leq i\leq r美元是某种功能,并表明如何获得美元A(x) ⁇ sum ⁇ k\ge)0}A_k(x)}A_k(x)}(x)(x)(x)(x) ⁇ sum ⁇ i=0 ⁇ k A_i)(x)美元=0}(x)美元=0 ⁇ k A_i)(x),美元=1美元(x),美元=1美元(x),美元=0\leq i i=leq r$,表明如何获得接近美元A(x) {sum *k\k\geq0}A_k(x)A_k(x)}(x)}(x)}(x)(我们将这一一般结果应用于列举的Dyckkkk(ckkkin、mozkin、sew Dyckkk) 和skew Dyckk) 和skin (ck) seq) seq(x) 路径的某些子段,根据第一个返回解解解解后,根据一个单一模式发生时以单状的单状的单状变变变变变变变变变异状态,根据一个单状最大变变变形变形变形变形变形变形变变变变变变变的频率的频率的出现最大波状,根据一个最大变形变形变形变形变形变形变形变形变形变形变形变形变形变形变形变形变形变形为$$=$$的出现达到$达到最大时,以。。。
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