



ABSTRACT OF THE DISSERTATION Multivariate Analogues of Catalan Numbers, Parking Functions, and their Extensions by Nicholas Anthony Loehr This document is concerned with the Catalan numbers and their generalizations. The Catalan numbers, which occur ubiquitously in combinatorics, are also connected to certain problems in representation theory, symmetric function theory, the theory of Macdonald polynomials, algebraic geometry, and Lie algebras. Garsia and Haiman introduced a bivariate analogue of the Catalan numbers, called the $q,t$Catalan sequence, in this setting. This sequence counts multiplicities of the sign character in a certain doubly graded $S_n$module called the diagonal harmonics module. Several classical $q$analogues of the Catalan numbers can be obtained from this sequence by suitable specializations. Haglund and Haiman separately proposed combinatorial interpretations for this $q,t$Catalan sequence by defining two statistics on Dyck paths. Garsia and Haglund later proved the correctness of these interpretations. Haglund, Haiman, and Loehr proposed similar statistics on labelled Dyck paths, which are conjectured to give the Hilbert series of the diagonal harmonics module.
 
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