No abstract

[15]. In fact, they formulated one conjecture for each pair (m, n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair (km, kn), with (m, n) co-prime and k > 1. Contents

This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine the theory of iterated Laurent series and a new algorithm for partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables. We discuss the efficiency of our algorithm by investigating problems studied by Andrews and his coauthors; our running time is much less than that of their Omega package.

We present a reduction algorithm that simultaneously extends Hermite's reduction for rational functions and the Hermite-like reduction for hyperexponential functions. It yields a unique additive decomposition and allows to decide hyperexponential integrability. Based on this reduction algorithm, we design a new method to compute minimal telescopers for bivariate hyperexponential functions. One of its main features is that it can avoid the costly computation of certificates. Its implementation outperforms Maple's function DEtools [Zeilberger]. Moreover, we derive an order bound on minimal telescopers, which is more general and tighter than the known one.

In a recent work Jim Haglund, Jennifer Morse and Mike Zabrocki proved a variety of identities involving Hall-Littlewood symmetric functions indexed by compositions. When they applied ∇ to these symmetric functions the resulting identities and computer data led them to some truly remarkable refinements of the shuffle conjecture. We prove here the symmetric function side of a recursion which combined with a recent parking function recursion of Angela Hicks [18] settles some some special cases of the Haglund-Morse-Zabrocki conjectures. Our main result of a compositional q, t-Catalan and Schröder theorem yields as a consequence surprisingly simple new proofs of the original q, t-Catalan and Schröder results.