give the following bijection that leads to a combinatorial interpretation of (\[cam\]).

\[cameron\] There is a bijection between partial $2$-Motzkin paths of length $i$ with $m$ marked R-visible up steps and $m+1$ elevation lines such that there is exactly one marked step between two adjacent elevation lines and free $3$-motzkin paths of length $i$ with exactly $m$ dotted horizontal steps.

Suppose that $P=P_1U^*P_2U^*\ldots P_{m}U^*P_{m+1}$ is a partial $2$-Motzkin path with $m$ marked R-visible up steps and $m+1$ elevation lines, we get a free $3$-motzkin path by changing all the marked up steps to dotted horizontal steps and applying the elevation operation for each $P_k$.

Conversely, given a free $3$-Motzkin path $P=P_1  \dashrightarrow P_2\dashrightarrow \cdots  P_{m}
\dashrightarrow P_{m+1}$ with $m$ dotted horizontal steps, where $\dashrightarrow$ denotes a dotted horizontal step, then we can get a partial $2$-motzkin path by changing each dotted horizontal step to a marked up step and the L-visible down steps of each $P_k$ to up steps by elevating their end points.

(138,30) (-0.15,0)[(1,1)[4]{}]{} (0.15,0)[(1,1)[4]{}]{}(4,4)(4.5,5)(5,4) (5,4)(5.5,3)(6,4) (6,4)(6.5,5)(7,4) (7,4)(7.5,3)(8,4) (8,4)[(1,1)[4]{}]{} (12,8)[(1,0)[4]{}]{} (16,8)[(1,-1)[4]{}]{} (20,4)[(1,1)[8]{}]{} (28,12)[(1,-1)[8]{}]{} (36,4)[(1,1)[4]{}]{} (39.85,8)[(1,1)[8]{}]{} (40.15,8)[(1,1)[8]{}]{}(42.5,9)[$*$]{} (40,8)[(1,1)[12]{}]{} (52,20)[(1,0)[4]{}]{} (0,4)(3,0)[20]{}[(1,0)[1]{}]{} (0,16)(3,0)[20]{}[(1,0)[1]{}]{} (0,0) (4,4) (8,4) (12,8) (16,8) (20,4) (24,8) (28,12) (32,8) (36,4) (40,8) (44,12) (48,16) (52,20) (56,20) (63,5)[$\longleftrightarrow$]{} (79.85,8)[(1,-1)[4]{}]{} (80.15,8)[(1,-1)[4]{}]{}(84,4)(84.5,5)(85,4) 