Automated analysis of combinatorial structures
Organizer and Chair: Stephen Melczer
(University of Waterloo and ENS Lyon)
- TORIN GREENWOOD, University of Pennsylvania
Asymptotics of the Coefficients of Bivariate Analytic Functions with Algebraic Singularities [PDF]
Flajolet and Odlyzko (1990) derived asymptotic formulae the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. In this talk, we use these multivariate analytic techniques to find asymptotic formulae for the coefficients of a broad class of bivariate generating functions with algebraic singularities.
- HUI HUANG, RISC - Linz
An Improved Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms [PDF]
The Abramov-Petkovsek reduction computes an additive decomposition of a
hypergeometric term, which extends the functionality of the Gosper
algorithm for indefinite hypergeometric summation. We improve the Abramov-Petkovsek
reduction so as to decompose a hypergeometric term as the sum of a summable
term and a non-summable one. The improved reduction does not solve any
auxiliary linear difference equation explicitly. It is also more efficient
than the original reduction according to computational experiments. Based
on this reduction, we design a new algorithm to compute minimal
telescopers for bivariate hypergeometric terms. The new algorithm can
avoid the costly computation of certificates.
- MARNI MISHNA, Simon Fraser University
A Baxter class of a different kind, and other walks on Young's Lattice [PDF]
Young's lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Walks in Young's lattice appear in a wide range of areas. We focus on the set of walks that start at the empty partition, and end at a row shape $\lambda=(m)$. Among the subfamilies we find a family of walks counted by Baxter numbers, and a surprising occurrence of Young Tableaux of bounded height. Our proofs are a mix of generating function analysis and explicit bijection. We can express our generating functions as diagonals of rational functions. Work in common with Burrill, Courtiel, Fusy, and Melczer.
- MARK WILSON, University of Auckland
Analytic Combinatorics in Several Variables [PDF]
I discuss the application of techniques in my 2013 book with Robin Pemantle to the asymptotic enumeration of lattice paths. I intend to show examples using the Sage package originally written by Alex Raichev. I hope to elicit comments from the audience about he best way to merge these techniques with more traditional univariate analysis.
- THOMAS WONG, University of British Columbia
Two Friendly Walks in a Sticky Slab [PDF]
In previous work we derived an exact solution of two friendly directed walks above a sticky wall with single and double interactions. This models the behaviour of a pair of attracting polymers above an adsorbing surface.
In this talk, we extend the model to two friendly directed walks in a sticky slab. I'll discuss the results we obtained for the single interaction model using a kernel method approach, and compare these results with similar models analysed previously. I'll also highlight some difficulties we encountered when extending to the double interaction model and the results obtained using a numerical approach.