A Positivity Result in the

We outline here a proof that a certain rational function C n (q; t) which has come to be known as the \q; t-Catalan" is in fact a polynomial with positive integer coeecients. Since C n (q; t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to nd a pair of statistics a; b on the collection D n of Dyck paths of length 2n yielding C n (q; t) = P t a(() q b((). Our proof is based on a recursion for C n (q; t) suggested by a pair of statistics recently proposed by J. Haglund. One of the byproducts of our results is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery At the 1988 Alghero meeting of the Lotharingian Seminar Macdonald introduced a 2-parameter symmetric function basis fJ X; q; t]g which has since proved to be fundamental in the Theory of Symmetric Functions. In recent years the Theory of Symmetric Functions has acquired particular importance because of its relation to the Representation Theory of Hecke algebras and the Symmetric Groups, and has been shown to have applicability in a wide range of scientiic and mathematical disciplines. In many of these developments the Macdonald polynomials and some of their specializations have played a central role. In the original paper 10] and in subsequent work ((1], 2], 3], 4], 6], 9]) a number of conjectures have been formulated which assert that certain rational functions in q; t are in fact polynomials with positive integer coeecients. For a decade these conjectures have resisted several various attempts of proof by a wide range of approaches. Although these conjectures lie squarely within the Theory of Symmetric Functions, the approaches range from diagonal actions of the symmetric group on polynomial rings in two sets of variables 1],,2], 4] to the Algebraic Geometry of Hilbert schemes 10]. EEorts to resolve these conjectures within the Theory of Symmetric Functions, have led to the discovery of a variety of new methods to deal with symmetric function identities 2], 3], 6], 7]. In this paper we outline an argument that yields a purely symmetric function proof of one of these conjectures. To state the result we need some deenitions and notational conventions. A partition will always be identiied with its …