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Singularly Perturbed Discrete Differential Equations and Pattern Counts in Simple Triangulations

Authors Michael Drmota , Eva-Maria Hainzl

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
Keywords
  • Discrete differential equations
  • catalytic equations
  • generating functions

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Abstract

Discrete differential equations of order k are of the form R(z,u,F(z,u),Δ F(z,u),…,Δ^kF(z,u)) = 0, where Δ F(z,u) = (F(z,u)-F(z,0))/u and Δ^k F(z,u) = Δ(Δ^{k-1} F(z,u)) for k ≥ 2. Such equations appear most prominently in planar map enumeration but also in several other contexts such as statistical mechanics, lattice path enumeration, pattern avoiding permutations or stack-sortable permutations. Mostly, one is interested in the function F(z,0) that is usually the corresponding counting generating function. In this work, we consider discrete differential equations with an additional parameter x, where the order of the equation is 1 for x = 1 but k > 1 for x ≠ 1. We call such equations singularly perturbed. The solution theory of higher order discrete differential equations is much more involved than for degree 1 and it is a priori not clear that there is a smooth transition from x = 1 to x ≠ 1. The main contribution of this work is to show that there is actually a smooth transition under certain natural assumptions. As an application of this result we consider pattern counts in triangular planar maps and derive a central limit theorem for these counts.

Cite As Get BibTex

Michael Drmota and Eva-Maria Hainzl. Singularly Perturbed Discrete Differential Equations and Pattern Counts in Simple Triangulations. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.17

Author Details

Michael Drmota
  • Institute of Discrete Mathematics and Geometry, TU Wien, Austria
Eva-Maria Hainzl
  • LIX, École polytechnique, CNRS, Institut Polytechnique de Paris, France

Funding

  • Drmota, Michael: This work was partially founded by the Austrian Science Foundation FWF, projects P 35016, F 100203.
  • Hainzl, Eva-Maria: This work was partially founded by the Austrian Science Foundation FWF, projects P 35016, F 100203 and ANR LOUCCOUM.

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