72 research outputs found

    A Fast Algorithm for MacMahon's Partition Analysis

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    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

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    Let ai,j(n)a_{i,j}(n) denote the number of walks in nn steps from (0,0)(0,0) to (i,j)(i,j), with steps (Β±1,0)(\pm 1,0) and (0,Β±1)(0,\pm 1), never touching a point (βˆ’k,0)(-k,0) with kβ‰₯0k\ge 0 after the starting point. \bous and Schaeffer conjectured a closed form for the number aβˆ’i,i(2n)a_{-i,i}(2n) when iβ‰₯1i\ge 1. In this paper, we prove their conjecture, and give a formula for aβˆ’i,i(2n)a_{-i,i}(2n) for iβ‰€βˆ’1i\le -1.Comment: 7 page

    A Residue Theorem for Malcev-Neumann Series

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    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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