[Paper Review] Counting With Irrational Tiles

Scott Garrabrant, Igor Pak|arXiv (Cornell University)|Jul 30, 2014

Advanced Combinatorial Mathematics36 references6 citations

TL;DR

This paper introduces and characterizes tile counting functions using irrational-length tiles in unit-height rectangles, proving they correspond exactly to diagonals of $ aturals$-rational generating functions and certain binomial multisums. The key contribution is a complete characterization of such sequences via multivariate rational generating functions and connections to hypergeometric functions and Catalan numbers.

ABSTRACT

We introduce and study the number of tilings of unit height rectangles with irrational tiles. We prove that the class of sequences of these numbers coincides with the class of diagonals of N-rational generating functions and a class of certain binomial multisums. We then give asymptotic applications and establish connections to hypergeometric functions and Catalan numbers.

Motivation & Objective

• To study the combinatorial structure of tiling sequences formed by irrational-length tiles in unit-height rectangles.
• To characterize the class of such tiling functions and relate them to known classes of generating functions.
• To establish connections between irrational tile tilings, diagonals of rational generating functions, and binomial multisums.
• To explore asymptotic behavior and links to special functions like hypergeometric functions and Catalan numbers.
• To investigate whether Catalan numbers can arise as asymptotic counts in irrational tile tilings, posing a conjecture on their non-representability.

Proposed method

• Define tile counting functions $f_{T,\varepsilon}(n)$ for fixed irrational tile sets $T$ and fixed offset $\ackslash varepsilon$, counting tilings of $[1\times(n+\varepsilon)]$.
• Use a generalized transfer-matrix method to show that such functions arise as diagonals of multivariate $ aturals$-rational generating functions.
• Characterize the class $\mathcal{F}$ of all such functions as precisely the diagonals of $\mathcal{R}_k$, the multivariate $ aturals$-rational functions.
• Establish that $f(n)$ is representable as a multivariate binomial sum $\sum_{\mathbf{v}} \prod_{i=1}^r \binom{a_{i1}v_1 + \cdots + a_{id}v_d + a'_i n + a''_i}{\cdots}$, linking to combinatorial multisums.
• Apply results from formal power series and hypergeometric function theory to analyze asymptotic behavior.
• Use cycle decomposition and multiplicity arguments inspired by combinatorial linear algebra and 'cycle popping' techniques to prove structural properties.

Experimental results

Research questions

• RQ1 Which sequences of integers can arise as the number of tilings of $[1\times(n+\varepsilon)]$ using a fixed finite set of irrational-length tiles?
• RQ2 Can every such tiling sequence be represented as a diagonal of a multivariate $ aturals$-rational generating function?
• RQ3 Is there a connection between irrational tile tilings and binomial multisums or hypergeometric functions?
• RQ4 Can the Catalan numbers $C_n$ appear as the asymptotic growth of such irrational tile tiling sequences?
• RQ5 Are there nonnegative rational generating functions that are not $ aturals$-rational, and how do they relate to this tiling framework?

Key findings

• The class of tile counting functions with irrational tiles coincides exactly with the class of diagonals of $ aturals$-rational generating functions.
• Every such tiling function $f(n)$ can be expressed as a multivariate binomial sum of the form $\sum_{\mathbf{v}} \prod_{i=1}^r \binom{a_{i1}v_1 + \cdots + a_{id}v_d + a'_i n + a''_i}{\cdots}$.
• The generating function for the number of tilings with two irrational tiles $[1\times(\frac{1}{2} \pm \alpha)]$ is $F(x) = \frac{1}{\sqrt{1-4x}}$, corresponding to the central binomial coefficients $\binom{2n}{n}$.
• The asymptotic growth of such tiling sequences is governed by the singularities of their generating functions, and in some cases, the constant term in the asymptotic expansion is known to be transcendental (e.g., involving $\pi$ and $\Gamma(1/4)$).
• The paper provides strong evidence—via hypergeometric function analysis—that the Catalan numbers $C_n$ cannot be asymptotically realized by any irrational tile tiling function.
• A conjecture is proposed that no irrational tile tiling function satisfies $f(n) \sim C_n$ as $n \to \infty$, based on the inability to produce the constant $\frac{\pi}{3\sqrt{3}}$ from the relevant hypergeometric products.