We use the Mansour-Vainshtein theory of kernel shapes [14] to decompose the set $S_n(1324)$ of 1324-avoiding permutations of length n into small pieces that are governed by some kernel shape λ. An enumeration of the set $K_{m,c}$ of all kernel shapes with length m and capacity c allows to express the generating function for the number of 1324-avoiding permutations of length n in terms of the $P_m(C(x))={\sum }_c|K_{m,c+1}|C^c(x)$ where $P_m(x)$ is a polynomial and $C(x)$ is the generating function for the Catalan numbers. This allows us to write down a systematic procedure for finding a lower bound for approximating the Stanley-Wilf limit of the pattern 1324. We use an implementation of this method in the OpenCL framework to compute such a bound explicitly.

我们使用核形状的Mansour-Vainshtein理论[14]分解集合 ${\mathrm{小号}}_{ñ}（1324）$避免将1324的长度n排列成小块，这些小块由某个核形状λ支配。集合的枚举${ķ}_{米，C}$所有内核形状，其具有长度的米和容量Ç允许表达对1324-避免长度的排列数生成函数Ñ中的条款$P_{米}（C（X））={\sum }_C|{ķ}_{米，C+\mathrm{1个}}|C^C（X）$ 在哪里 $P_{米}（X）$ 是一个多项式， $C（X）$是加泰罗尼亚语数字的生成函数。这使我们能够写下系统的过程，以找到下界以近似于模式1324的Stanley-Wilf极限。我们在OpenCL框架中使用此方法的实现来显式计算该边界。