Limits of modified higher q, t-Catalan numbers

作者：Lee Kyungyong^*; Li Li; Loehr Nicholas A

来源：ELECTRONIC JOURNAL OF COMBINATORICS, 2013, 20(3): P4.

DOI：10.37236/3201

The q, t-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher q, t-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of several versions of the modified higher q, t-Catalan numbers and show that these limits equal the generating function for integer partitions. We also identify certain coefficients of the higher q, t-Catalan numbers as enumerating suitable integer partitions, and we make some conjectures on the homological significance of the Bergeron-Garsia nabla operator.

• 出版日期2013-7-19
• 单位美国弗吉尼亚理工大学(Virginia Tech)