Generalized Narayana polynomials and their $q$-analogues

Generating polynomials of the statistics $\mathrm{rise}$, $\mathrm{des}$ and $\mathrm{inv}$ are considered on the entered $312$-avoiding GS-permutations of an order $r\geq1$. It is shown that the polynomials of the statistics $\mathrm{rise}$ and $\mathrm{des}$ are some generalizations of the known Narayana polynomials. For the generalized Narayana polynomials, the inverse generating function, an algebraic equation for the generating function and a recursion relation with multiple convolutions are obtained. For the generating polynomials of pair $\mathrm{(des,inv)}$, an analogue of the obtained recursion relation and an equation for the generating function of these polynomials are found. Their particular case leads to the corresponding $q$-analogues of generalized Narayana polynomials.