Let $ P$ be a subset of $\mathbb{N}^2\setminus {(0,0)}$ with the two following conditions:
$(1) \quad $ every $x \in P$ has the form $(n,n)$, $(n+1,n)$ or $(n,n+1)$, and
$(2) \quad (n+1,n) \in P$ if and only if $(n,n+1) \in P$.
Let $P_m$ denote the set of paths from the origin to $(m,m)$ using steps from $P$, that consist of the single step (m,m) and also stay strictly below the diagonal ($y=x$) except at the origin and $(m,m)$.
I am some problem understanding how the elements of $P_m$ looks like. In particular, let us work with $P_3$. Let y be path from $(0,0) \rightarrow (2,1) \rightarrow (3,3)$. Is $y \in P_3$ ? I would really appreciate it if some can please explain this to me. Example of an element in $P_3$ would be great or perhaps a general explanation.