In this paper, on one hand, we propose a new type of symmetric function to interpret the bi$^{s}$nomial coefficients and their analogs. On other hand, according to this function, we give an interpretation of these coefficients by lattice paths and tiling. Some identities of these coefficients are also established. This work is an extension of the results of Belbachir and Benmezai's ''A $\mathit{q}$-analogue for bi$^{\mathit{s}}$nomial coefficients and generalized Fibonacci sequences".