(70,19) (74,15)(78,19) (82,15) (86,19) (90,15)

(93,19)[$\longleftrightarrow$]{}

(0,3)[$23$]{} (5.3,0.3)[(1,1)[8]{}]{} (13,8)[(1,-1)[7.7]{}]{} (18.5,1.9)[$\ast$]{}(21.3,0.3)[(1,1)[4]{}]{} (25,4)[(1,-1)[3.7]{}]{} (5,0) (9,4) (13,8) (17,4) (21,0) (25,4) (29,0)

(32,4)[$\longleftrightarrow$]{}

(43,3)[$234$]{} (48,0)[(1,1)[4]{}]{} (52,4)[(1,-1)[3.7]{}]{} (53.5,1.9)[$\ast$]{}(56.3,0.3)[(1,1)[4]{}]{} (60,4)[(1,-1)[3.7]{}]{} (48,0) (52,4) (56,0) (60,4) (64,0)

(67,4)[$\longleftrightarrow$]{}

(78,3)[$2341$]{} (85,0)[(1,1)[4]{}]{} (89,4)[(1,-1)[3.7]{}]{} (90.5,1.9)[$\ast$]{} (85,0) (89,4) (93,0)

(96,4)[$\longleftrightarrow$]{} (107,3)[$2341$]{}

Weighted Partial Motzkin Paths 
===============================

A [*Motzkin path*]{} of length $n$ is a path going from $(0,0)$ to $(n,0)$ consisting of up steps $U=(1,1)$, down steps $D=(1,-1)$ and horizontal steps $H=(1,0)$, which never goes below the $x$-axis. A [*$(k,t)$-Motzkin path*]{} is a Motzkin path such that each horizontal step is weighted by $k$, each down step is weighted by $t$ and each up step is weighted by $1$. The case $k=2, t=1$ gives the $2$-Motzkin paths which have been introduced by Barcucci, del Lungo, Pergola and Pinzani [@BdLPP] and have been studied by Deutsch and Shapiro [@deutschs]. The [*weight* ]{} of a path is the product of the weights of all its steps. Denote by $|P|$ the weight of a path $P$. The [*weight* ]{} of a set of paths is the sum of the total weights of all the paths. For any step, we say that it is at level $k$ if the $y$-coordinate of its end point is at 