Limits of Modified Higher (q,t)-Catalan Numbers

Details Limits of Modified Higher (q,t)-Catalan Numbers Limits of Modified Higher (q,t)-Catalan Numbers

2011-10-26

arxiv.org/abs/1110.5850v1

Mathematics

Combinatorics

19 pages, comments welcome

Scientific paper

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. 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. This result supports the conjecture that the combinatorial definitions of higher q,t-Catalan numbers are indeed equivalent to the algebraic and geometric definitions.

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