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 tau-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 tau-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 WEulerian polynomials andWNarayana 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.