132-避免排列中的象限标记网格模式II
Quadrant marked mesh patterns in 132-avoiding permutations II
摘要：
给定对称群$S_n$中的置换$\sg = \sg_1...\sg_n$，如果至少存在$a$，则我们说$\sg_i$与$\sg$中标记的网格模式$MMP(a,b,c,d)$匹配ABMR指向$\sg$中大于$\sg_i$的$\sg_i$的右侧，至少$b$指向$\sg$中的$\sg_i$的左侧，并且大于$\sg_i$，至少$c$指向$\sg$中小于$\sg_i$的$\sg_i$的左侧，至少$d$指向$\sg$中小于$\sg_i$的$\sg_i$的右侧。本文是从\cite{kitremtie}开始的对132-避排列中象限标记网格模式分布的系统研究的继续，我们主要研究了$a,b,c,d$中恰好有一个大于零而其余元素为零的132-避排列中$MMP(a,b,c,d)$的匹配数分布.本文研究了$MMP(a,b,c,d)$在132-避免排列中匹配数的分布，其中恰好有两个$a,b,c,d$大于零，其余元素为零.我们提供明确的递归关系，列举我们的对象，可以用来给封闭的形式与这种分布的生成函数。在许多情况下，我们提供组合解释的系数，出现在我们的生成函数。将在\cite{kitremtieIII}中研究象限标记网格模式$MMP(a,b,c,d)$的情况，其中三个或更多$a,b,c,d$被约束为大于0。
英文摘要：
Given a permutation $\sg = \sg_1...\sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the marked mesh pattern $MMP(a,b,c,d)$ in $\sg$ if there are at least $a$ points to the right of $\sg_i$ in $\sg$ which are greater than $\sg_i$, at least $b$ points to the left of $\sg_i$ in $\sg$ which are greater than $\sg_i$, at least $c$ points to the left of $\sg_i$ in $\sg$ which are smaller than $\sg_i$, and at least $d$ points to the right of $\sg_i$ in $\sg$ which are smaller than $\sg_i$. This paper is continuation of the systematic study of the distribution of quadrant marked mesh patterns in 132-avoiding permutations started in \cite{kitremtie} where we mainly studied the distribution of the number of matches of $MMP(a,b,c,d)$ in 132-avoiding permutations where exactly one of $a,b,c,d$ is greater than zero and the remaining elements are zero. In this paper, we study the distribution of the number of matches of $MMP(a,b,c,d)$ in 132-avoiding permutations where exactly two of $a,b,c,d$ are greater than zero and the remaining elements are zero. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions. The case of quadrant marked mesh patterns $MMP(a,b,c,d)$ where three or more of $a,b,c,d$ are constrained to be greater than 0 will be studied in \cite{kitremtieIII}.
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