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We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.
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q,t$q,t$‐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.
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- Award ID(s):
- 1945212
- PAR ID:
- 10508996
- Publisher / Repository:
- London Math. Soc.
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 56
- Issue:
- 3
- ISSN:
- 0024-6093
- Page Range / eLocation ID:
- 1207 to 1226
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/blms.12989
