72 research outputs found
A Fast Algorithm for MacMahon's Partition Analysis
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.Comment: 22 page
Proof of a Conjecture on the Slit Plane Problem
Let denote the number of walks in steps from to
, with steps and , never touching a point
with after the starting point. \bous and Schaeffer
conjectured a closed form for the number when . In this
paper, we prove their conjecture, and give a formula for for
.Comment: 7 page
A Residue Theorem for Malcev-Neumann Series
In this paper, we establish a residue theorem for Malcev-Neumann series that
requires few constraints, and includes previously known combinatorial residue
theorems as special cases. Our residue theorem identifies the residues of two
formal series that are related by a change of variables. We obtain simple
conditions for when a change of variables is possible, and find that the two
related formal series in fact belong to two different fields of Malcev-Neumann
series. The multivariate Lagrange inversion formula is easily derived and
Dyson's conjecture is given a new proof and generalized.Comment: 22 pages, extensive revisio
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