Generalized narayana polynomials and their q-analogues

Generating polynomials of the statistics rise, des and inv are considered on the entered 312-avoiding GS-permutations of an order r ^ 1. It is shown that the polynomials of the statistics rise and 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 (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.

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