Mathematics Colloquia and Seminars

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The noncrossing set partitions of {1, 2, ... , n} are famously counted by the Catalan number $\frac{1}{2n+1} {2n \choose n}$. For any reflection group $W$, one can associate a `$W$-Catalan number' Cat($W$) and prove that it counts a $W$-analog of noncrossing partitions. Unfortunately, the only known proofs of this fact use the Cartan-Killing classification of reflection groups. In joint work with Armstrong and Reiner, I defined a new class of objects attached to $W$ called `$W$-noncrossing parking functions' and made a conjecture about them which uniformly implies the $W$-Catalan enumeration. I will explain this conjecture and give some recent evidence for it, including its proof in the (surprisingly) most challenging classical case -- that of the symmetric group.