Diagonal Asymptotics for Symmetric Rational Functions via ACSV

Authors Yuliy Baryshnikov, Stephen Melczer , Robin Pemantle, Armin Straub

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Yuliy Baryshnikov
  • University of Illinois, Department of Mathematics, 273 Altgeld Hall 1409 W. Green Street (MC-382), Urbana, IL 61801, USA
Stephen Melczer
  • University of Pennsylvania, Department of Mathematics, 209 South 33rd Street, Philadelphia, PA 19104, USA
Robin Pemantle
  • University of Pennsylvania, Department of Mathematics, 209 South 33rd Street, Philadelphia, PA 19104, USA
Armin Straub
  • University of South Alabama, Department of Mathematics and Statistics, 411 University Blvd N, MSPB 325, Mobile, AL 36688, USA

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Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle, and Armin Straub. Diagonal Asymptotics for Symmetric Rational Functions via ACSV. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We consider asymptotics of power series coefficients of rational functions of the form 1/Q where Q is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diagonal asymptotics across entire classes of functions, for example the general 3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the denominator in terms of elementary symmetric functions is 1 - e_1 + c e_d in any number d of variables. The ACSV analysis also explains a discontinuous drop in exponential growth rate for the GRZ class at the parameter value c = (d-1)^{d-1}, previously observed for d=4 only by separately computing diagonal recurrences for critical and noncritical values of c.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Analytic combinatorics
  • generating function
  • coefficient
  • lacuna
  • positivity
  • Morse theory
  • D-finite
  • smooth point


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