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.
@article{10_37236_3201,
author = {Kyungyong Lee and Li Li and Nicholas A Loehr},
title = {Limits of modified higher {\(q,t\)-Catalan} numbers},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {3},
doi = {10.37236/3201},
zbl = {1295.05268},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3201/}
}
TY - JOUR
AU - Kyungyong Lee
AU - Li Li
AU - Nicholas A Loehr
TI - Limits of modified higher \(q,t\)-Catalan numbers
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/3201/
DO - 10.37236/3201
ID - 10_37236_3201
ER -
%0 Journal Article
%A Kyungyong Lee
%A Li Li
%A Nicholas A Loehr
%T Limits of modified higher \(q,t\)-Catalan numbers
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/3201/
%R 10.37236/3201
%F 10_37236_3201
Kyungyong Lee; Li Li; Nicholas A Loehr. Limits of modified higher \(q,t\)-Catalan numbers. The electronic journal of combinatorics, Tome 20 (2013) no. 3. doi: 10.37236/3201
