中央研究院數學研究所

Taipei Postdoc Seminar

Symmetric Functions, Domino Tilings, and the Nabla operator

2025/12/17 (Wed.) 14:00~15:00

Date : 2025/12/17 (Wed.) 14:00~15:00
Location : Room 515, Cosmology Building, NTU
Speaker : Yi-Lin Lee (National Taiwan Normal University)
Organizer : Wille Liu (AS), Harry Richman (NCTS)
Abstract : The study of Macdonald symmetric polynomials has produced many interesting combinatorial objects. Perhaps the most famous and well-studied such objects are the q, t-Catalan numbers, which can be defined combinatorially as the sum over Dyck paths weighted by the area and dinv statistics. Numerous generalizations of the q, t-Catalan numbers have been developed, including extensions to Schroder paths and to nested families of Dyck paths. All of these objects have natural interpretations in terms of the nabla operator on symmetric functions.
In this talk, I will introduce the algebraic and combinatorial background of the nabla operator, and present its new connections with domino tilings of a certain region on the square lattice. In particular, a product formula for the q, t-generalization of domino tilings of the Aztec diamond, together with a combinatorial proof of the joint symmetry of the area and dinv statistics on the Aztec diamond, is presented. If time permits, I will also outline some proof ideas and related results on tiling enumeration. This talk does not assume any prior background.