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q,t-Catalan measures

Ian Cavey

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.

q,t-Catalan measures

Abstract

We introduce the -Catalan measures, a sequence of piece-wise polynomial measures on . 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 -Catalan numbers. Our main result realizes the -Catalan measures as a limit of higher -Catalan numbers as . We also give a geometric interpretation of the -Catalan measures. They are the Duistermaat-Heckman measures of the punctual Hilbert schemes parametrizing subschemes of supported at the origin.
Paper Structure (15 sections, 17 theorems, 50 equations, 3 figures, 1 table)

This paper contains 15 sections, 17 theorems, 50 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Higher $q,t$-Catalan Numbers
  3. Continuous Dyck Paths and the $q,t$-Catalan Measures
  4. Continuous Dyck Path Statistics
  5. Area and Bounce Polytopes
  6. $q,t$-Catalan Measures
  7. A Measure Preserving Transformation on Continuous Dyck Paths
  8. $m$-Dyck Path Combinatorics as $m\to\infty$
  9. Area and Dinv as $m\to\infty$
  10. $m$-Bounce Paths as $m\to\infty$
  11. The Bijections $\phi_m:\mathcal{D}^{(m)}_n\to\mathcal{D}^{(m)}_n$ as $m\to\infty$
  12. Higher $q,t$-Catalan Numbers as $m\to\infty$
  13. Connection to Hilbert Schemes
  14. Hilbert Schemes and Higher $q,t$-Catalan Numbers
  15. Duistermaat-Heckman Measure

Key Result

Theorem 1.1

For all $n\geq 1$, the $q,t$-Catalan measure $\mu_n$ is equal to the weak limit of measures on $\mathbb R^2$,

Figures (3)

  • Figure 1: Discrete density functions of higher $q,t$-Catalan numbers
  • Figure 2: An $m$-Dyck path and its corresponding continuous Dyck path
  • Figure 3: A continuous Dyck path of height $4$

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 2.1: Mellit M
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 17 more