Enumerative combinatorics
A Remarkable q, t-Catalan Sequence and q-Lagrange Inversion
Journal of Algebraic Combinatorics: An International Journal
The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
A proof of the q, t-Catalan positivity conjecture
Discrete Mathematics
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The Catalan numbers occur ubiquitously in combinatorics. R. Stanley's book Enumerative Combinatorics 2 (1999) and its addendum (http://www-math.mit.edu/~rstan/ec/catadd.pdf) list over 95 collections of objects counted by the Catalan numbers. We augment this list with two additional collections of permutations that are enumerated by the Catalan numbers. Furthermore, we show that the generating function for either collection, relative to the classical coinversion and major index statistics, is precisely the q, t-Catalan sequence of Garsia and Haiman. This is proved by exhibiting weight-preserving bijections between the given collections and the set of Dyck paths. The bijections are based on encodings of Dyck paths and permutations as sequences of partitions.