q,t-Catalan measures

TL;DR

This paper introduces the $q,t$-Catalan measures, continuous analogs of $q,t$-Catalan numbers, with geometric interpretation as Duistermaat-Heckman measures related to Hilbert schemes.

Contribution

It defines the $q,t$-Catalan measures, shows they are limits of higher $q,t$-Catalan numbers, and provides a geometric interpretation via Hilbert schemes.

Findings

$q,t$-Catalan measures are piece-wise polynomial measures on $ eal^2$.

They are limits of higher $q,t$-Catalan numbers as $m o \infty$.

The measures correspond to Duistermaat-Heckman measures of punctual Hilbert schemes.

Abstract

We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial measures on $\mathbb{R}^2$. These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the $q,t$-Catalan numbers. Our main result realizes the $q,t$-Catalan measures as a limit of higher $q,t$-Catalan numbers $C^{(m)}_n(q,t)$ as $m\to\infty$. We also give a geometric interpretation of the $q,t$-Catalan measures. They are the Duistermaat-Heckman measures of the punctual Hilbert schemes parametrizing subschemes of $\mathbb{C}^2$ supported at the origin.