Parallelogram Polyominoes and (Surprise!)-- The Diagonal HarmonicsAlgebra & Discrete Mathematics
|Speaker:||Angela Hicks, Stanford|
|Start time:||Mon, Sep 30 2013, 11:00AM|
A recent paper of Dukes and Le Borgne studied two statistics on parallelogram polyominoes-- two nonintersecting paths, each composed of north and east steps and bounded by a rectangular m x n bounding box. Conjecturing that q,t-counting the polyominoes by the two statistics resulted in polynomials that were symmetric in q and t as well as m and n, they called the statistics "area"' and "bounce," in reference to the historic statistics on parking functions. This talk will discuss a following joint paper (with the original two authors, Aval, and D'Adderio) which introduced a third statistic, "dinv," on polyominoes and demonstrated the conjectured symmetries. In a surprising twist, the proof illuminates a direct link from polyominoes to parking functions and the famous space of diagonal harmonics. This talk will assume only a basic knowledge of linear algebra and combinatorics; it will introduce the history of problem and its surprising conclusion.