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Abstract

This chapter contains an account of a two-parameter version of the Catalan numbers, and corresponding two-parameter versions of related objects such as parking functions and Schröder paths, which have become important in algebraic combinatorics and other areas of mathematics as well. Although the original motivation for the definition of these objects was the study of Macdonald polynomials and the representation theory of diagonal harmonics, in this account we focus only on the combinatorics associated to their description in terms of lattice paths. Hence this chapter can be read by anyone with a modest background in combinatorics. In Section 1 we include basic facts involving q-analogues, permutation statistics, and symmetric functions which we need in later sections. Sections 2, 3, and 4 contain the results on the q, t-versions of the Catalan numbers, parking functions, and Schröder paths, respectively. Section 5 contains a brief account of the recent exciting extensions of these objects which have arisen in the study of string theory, knot invariants, and the Hilbert scheme from algebraic geometry.

Flow Polytopes and the Space of Diagonal Harmonics

“…For more background on DH n and the Shuffle Conjecture see [2,12,13]. For the definition of area and dinv on parking functions see [16, §1, §2].…”

Area-width scaling in generalised Motzkin paths

“…This model has been studied previously in the combinatorics literature with a focus on bijections [17]. Motzkin and Schöder paths are included in this family of walks and correspond to the cases = 1 and 2, respectively, whereas Dyck paths can be identified with the limiting case = ∞ [18].…”