**The Combinatorics of the q,t-Catalan**

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Greg Warrington

University of Vermont

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**Monday, September 30th, 4:00 PM**

**Votey 207**

**Abstract**

The Catalan numbers form a fundamental integer sequence with over 200 different interpretations. Generalizing these, the q,t-Catalan numbers are two-variable analogues that arise in the study of a certain representation of the symmetric group of permutations. Their combinatorics is very rich. This talk will begin by defining the polynomials and exploring, from an elementary point of view, some of the combinatorics associated to them. Included in the discussion will be one shockingly elementary fact for which there is, as of yet, no combinatorial proof :(. We will finish by sketching some of the algebra and representation theory that motivates the study of them.