Symmetric functions, tilings, and the nabla operator
Release Date： 2025-11-12 Unit： Department of Mathematics
Professor：Dr. Yi-Lin Lee (National Taiwan Normal University)
Theme：Symmetric functions, tilings, and the nabla operator
Date：Nov. 27th (Thu) 15:30 ~ 17:00
Location：Hong-Jing Building, M107
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 $\nabla$ 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.
Theme：Symmetric functions, tilings, and the nabla operator
Date：Nov. 27th (Thu) 15:30 ~ 17:00
Location：Hong-Jing Building, M107
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 $\nabla$ 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.
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Updated: 2025-11-12
Category: Speech
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