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Introducing irrational enumeration: analytic combinatorics for objects of irrational size

David Bevan, Julien Condé

TL;DR

The paper extends analytic combinatorics to classes with irrational object sizes by introducing Ribenboim generating functions and their exponential transforms to generalized Dirichlet series. It proves a principal result for primitive irrational classes: if the IGF has a dominant singularity at radius ρ and can be decomposed as f(z)=g(z)+h(z)/(1−z/ρ)^α, then the cumulative count |C_{≤x}| has the precise asymptotic form with a log(1/ρ) in the denominator, paralleling classical results for integer sizes. The authors develop broad, constructive criteria for primitive irrationality and apply the theory to a wide array of problems—tilings, lattice paths, and plane trees—delivering explicit asymptotics, phase-transition phenomena, and structural insights, while also addressing rational and shifted-rational classes and highlighting open questions in off-diagonal and higher-term analyses. The work bridges irrational size enumeration with known Dirichlet-series and Tauberian machinery, revealing intricate connections to entropy-like expressions and periodic oscillations in non-primitive cases. Overall, the framework significantly broadens the reach of analytic combinatorics to non-integer sizes, enabling precise asymptotics and qualitative phase behavior across diverse combinatorial models.

Abstract

We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. We illustrate the technique with a variety of applications, including tilings with tiles of irrational area, ordered integer factorizations, lattice walks enumerated by Euclidean length, and plane trees with vertices of irrational size. We also explore phase transitions in the asymptotics of families of irrational combinatorial classes.

Introducing irrational enumeration: analytic combinatorics for objects of irrational size

TL;DR

The paper extends analytic combinatorics to classes with irrational object sizes by introducing Ribenboim generating functions and their exponential transforms to generalized Dirichlet series. It proves a principal result for primitive irrational classes: if the IGF has a dominant singularity at radius ρ and can be decomposed as f(z)=g(z)+h(z)/(1−z/ρ)^α, then the cumulative count |C_{≤x}| has the precise asymptotic form with a log(1/ρ) in the denominator, paralleling classical results for integer sizes. The authors develop broad, constructive criteria for primitive irrationality and apply the theory to a wide array of problems—tilings, lattice paths, and plane trees—delivering explicit asymptotics, phase-transition phenomena, and structural insights, while also addressing rational and shifted-rational classes and highlighting open questions in off-diagonal and higher-term analyses. The work bridges irrational size enumeration with known Dirichlet-series and Tauberian machinery, revealing intricate connections to entropy-like expressions and periodic oscillations in non-primitive cases. Overall, the framework significantly broadens the reach of analytic combinatorics to non-integer sizes, enabling precise asymptotics and qualitative phase behavior across diverse combinatorial models.

Abstract

We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. We illustrate the technique with a variety of applications, including tilings with tiles of irrational area, ordered integer factorizations, lattice walks enumerated by Euclidean length, and plane trees with vertices of irrational size. We also explore phase transitions in the asymptotics of families of irrational combinatorial classes.

Paper Structure

This paper contains 24 sections, 10 theorems, 118 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Establishing primitive irrationality
  4. Tiling and packing
  5. Strips of width 1
  6. Tilings with a final partial tile
  7. Maximal packings
  8. The structure of a typical tiling
  9. Ordered integer factorizations
  10. Infinite sets of rational tiles
  11. Adjacency constraints
  12. Irrational tilings that are not primitive
  13. Floor tiling
  14. Lattice walks and trees
  15. Motzkin paths
  16. ...and 9 more sections

Key Result

Theorem 1

Suppose that $\mathcal{C}$ is a primitive irrational combinatorial class. If its irrational generating function $f_\mathcal{C}(z)$ has positive radius of convergence $\rho<1$, and one can write where $\alpha\notin-\mathbb{N}$, and both $g$ and $h$ are analytic on the cut disk $\overline{D}(0,\rho) \setminus \mathbb{R}^{\leqslant0}$, then the asymptotic number of objects in $\mathcal{C}$ of size a

Figures (14)

  • Figure 1: A tiling of a strip using seven tiles of length $1$ and four of length $\sqrt2$
  • Figure 2: The mapping of the region of convergence under the exponential transform
  • Figure 3: A tiling with a final partial tile of a strip of length $\pi^2$, using square tiles and tiles of length $\sqrt2$
  • Figure 4: A maximal packing of a strip of length $\pi^2$, using square tiles and tiles of length $\sqrt2$
  • Figure 5: A tiling in $\mathcal{W}^{(\sqrt2)}$
  • ...and 9 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Conjecture 6
  • Proposition 8: HR1915
  • ...and 7 more