Dimensions of τ-tilting modules over path algebras and preprojective algebras of Dynkin type

In this paper, we introduce a new generating function called d-polynomial for the dimensions of τ-tilting modules over a given finite dimensional algebra. Firstly, we study basic properties of d-polynomials and show that it can be realized as a certain sum of the f-polynomials of the simplicial complexes arising from τ-rigid pairs. Secondly, we give explicit formulas of d-polynomials for preprojective algebras and path algebras of Dynkin quivers by using a close relation with W-Eulerian polynomials and W-Narayana polynomials. Thirdly, we consider the ordinary and exponential generating functions defined from d-polynomials and give closed-form expressions in the case of preprojective algebras and path algebras of Dynkin type A.