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.

中文翻译：

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q,t-加泰罗尼亚措施

我们介绍-Catalan 测度，一系列分段多项式测度。这些度量是根据平面上连续路径族的合适面积、dinv 和弹跳统计来定义的，并且与- 加泰罗尼亚语号码。我们的主要结果实现了-加泰罗尼亚措施作为更高的限制- 加泰罗尼亚语号码作为。我们还给出了几何解释-加泰罗尼亚措施。它们是准时希尔伯特方案的 Duistermaat-Heckman 测度，参数化子方案在原点得到支持。