(110,80)[(1,-1)[20]{}]{} (130,60)[(1,-1)[20]{}]{} (150,40)[(1,-1)[20]{}]{} (170,20)[(1,1)[20]{}]{} (8,9)[0]{} (28,9)[1]{} (48,9)[2]{} (68,9)[3]{} (88,9)[4]{} (108,9)[5]{} (128,9)[6]{} (148,9)[7]{} (168,9)[8]{} (188,9)[9]{} (-3,17)[1]{} (-3,37)[2]{} (-3,57)[3]{} (-3,77)[4]{} (10,20)[(1,0)[180]{}]{} (10,80)[(1,0)[180]{}]{} (10,20)[(0,1)[60]{}]{} (190,20)[(0,1)[60]{}]{} (10,60) (30,80) (50,60) (70,40) (90,60) (110,80) (130,60) (150,40) (170,20) (190,40)

Let us now consider the following set of paths. Let $p=(p_1,p_2,\ldots,p_L)$ be a sequence of $1$’s and $2$’s and let ${\lambda}=({\lambda}_1,{\lambda}_2)$ be the content of $p$ where ${\lambda}_i$ is the number of $i$’s in $p$. Denote by ${\mathcal{P}}_{L,{\lambda}}$ the set of all $p=(p_1,\ldots,p_L)$ with content ${\lambda}$. Another way to represent $p$ is by height variables $\sigma=(\sigma_0,\sigma_1,\ldots,\sigma_L)$ where by convention $\sigma_0=3$ and $\sigma_i=\sigma_{i-1}+1$ if $p_i=1$ and $\sigma_i=\sigma_{i-1}-1$ if $p_i=2$ for $1\le i\le L$. We will use $\sigma$ and $p$ interchangeably. We are actually mostly concerned here with the set ${\mathcal{P}}_L:={\mathcal{P}}_{L,{\lambda}}$ where ${\lambda}=(\lfloor \frac{L}{2}\rfloor,\lfloor \frac{L+1}{2}\rfloor)$.

Consider the following subsets of ${\mathcal{P}}_L$. Let ${\overline{\mathcal{P}}}_L$ be the subset of ${\mathcal{P}}_L$ consisting of all paths in the strip of height four, that is all $\sigma\in{\mathcal{P}}_L$ such that $1\le \sigma_i\le 4$ for all $1\le i\le L$. Furthermore, let ${\mathcal{P}}_L^{\uparrow,j}$ be the set of all paths $\sigma\in{\mathcal{P}}_L$ such that there exist indices $1\le i_1<i_2<\cdots<i_j\le L$ so that $\sigma_{i_1},\sigma_{i_3},\sigma_{i_5},\ldots$ are greater than four and $\sigma_{i_2},\sigma_{i_4},\sigma_{i_6},\ldots$ are less than one. Similarly let ${\mathcal{P}}_L^{\downarrow,j}$ be the set of 