of the elevation of a partial Motzkin path with respect to the line $y=2$.

(150,30) (-0.15,0)[(1,1)[4]{}]{} (0.15,0)[(1,1)[4]{}]{}(4,4)[(1,0)[4]{}]{} (8,4)[(1,1)[4]{}]{} (12,8)[(1,-1)[4]{}]{} (16,4)[(1,0)[4]{}]{} (20,4)[(1,1)[8]{}]{} (28,12)[(1,-1)[8]{}]{} (35.85,4)[(1,1)[4]{}]{} (36.15,4)[(1,1)[4]{}]{}(40,8)[(1,1)[8]{}]{}(48,16)[(1,-1)[4]{}]{} (52,12)[(1,0)[4]{}]{} (0,8)(3,0)[20]{}[(1,0)[1]{}]{}

(0,0) (4,4) (8,4) (12,8) (16,4) (20,4) (24,8) (28,12) (32,8) (36,4) (40,8) (44,12) (48,16) (52,12) (56,12)

(63,5)[$\longleftrightarrow$]{}

(79.85,8)[(1,-1)[4]{}]{} (80.15,8)[(1,-1)[4]{}]{}(84,4)[(1,0)[4]{}]{} (88,4)[(1,1)[4]{}]{} (92,8)[(1,-1)[4]{}]{} (96,4)[(1,0)[4]{}]{} (100,4)[(1,1)[8]{}]{} (108,12)[(1,-1)[8]{}]{} (115.85,4)[(1,-1)[4]{}]{} (116.15,4)[(1,-1)[4]{}]{}(120,0)[(1,1)[8]{}]{} (128,8)[(1,-1)[4]{}]{} (132,4)[(1,0)[4]{}]{}

(80,8) (84,4) (88,4) (92,8) (96,4) (100,4) (104,8) (108,12) (112,8) (116,4) (120,0) (124,4) (128,8) (132,4) (136,4)

We now introduce the notion of [*free Motzkin* ]{} paths which are lattice paths starting from $(0,0)$ and using up steps $U=(1,1)$, down steps $D=(1,-1)$ and horizontal steps $H=(1,0)$. Note that there is no restriction so that the paths may go below the $x$-axis. A free $(k,t)$-Motzkin path is a free Motzkin path in which the steps are weighted in the same way as for $(k,t)$-Motzkin paths, namely, an up step has weight one, a horizontal step has weight $k$ and a down step has weight $t$.

For a free Motzkin path $P$ we may analogously define the [*L-visible*]{} down steps as the down steps that are visible from the far left. It is clear that a complete Motzkin path (a partial Motzkin path with ending point on the $x$-axis) has no 