[Paper Review] The area generating function for simplex-duplex polyominoes

Svjetlan Feretić|ArXiv.org|Oct 26, 2009

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper introduces simplex-duplex polyominoes—column-duplex polyominoes where no two adjacent columns both have two connected components—and derives their area generating function using an enhanced version of the Temperley method. The method involves adding one or two columns at a time and solving a functional equation involving both first and second derivatives of the generating function, yielding an asymptotic growth rate of approximately 0.119443 × 3.522020^n for n-celled simplex-duplex polyominoes.

ABSTRACT

Back in the early days of polyomino enumeration, a model called column-convex polyominoes was introduced and its area generating function was found. That generating function is rational: the numerator has degree four and the denominator has degree three. Let a column-duplex polyomino be a polyomino whose columns can have either one or two connected components. A simplex-duplex polyomino is a column-duplex polyomino in which there is no occurrence of two adjacent columns each having two connected components. Simplex-duplex polyominoes are not easy to deal with, but their area generating function can still be found. To find this generating function, we use an upgraded version of the Temperley method. Though that technique is widely used in these times, our application presents two interesting features. Firstly, we add one or two columns at a time, thus bypassing those simplex-duplex polyominoes which end with a two-component column. (It is somewhat more usual to add just one column at a time.) Secondly, we obtain a functional equation that involves both the first and the second derivatives of the sought-for generating function. (Such equations usually involve the first derivative only. In some cases, no derivative is involved at all.) Right because of this latter interesting feature, the Temperley method produces a very complicated formula for the generating function. Anyway, from that formula it is easy to compute Taylor polynomials. Thus we get plenty of evidence that the number of n-celled simplex-duplex polyominoes behaves asymptotically as 0.119443*3.522020^n. For comparison, the number of n-celled column-convex polyominoes behaves asymptotically as 0.180916*3.205569^n.

Motivation & Objective

• To develop a solvable model that interpolates between column-convex and column-duplex polyominoes, which are otherwise intractable for exact enumeration.
• To define and analyze simplex-duplex polyominoes, a subclass of column-duplex polyominoes where no two adjacent columns both have two connected components.
• To derive the bivariate generating function G(q,w) counting simplex-duplex polyominoes by area (q) and number of two-component columns (w).
• To compute the area generating function G(q,1) and determine its asymptotic behavior for the total number of n-celled simplex-duplex polyominoes.

Proposed method

• An upgraded version of the Temperley method is employed, which adds one or two columns at a time to avoid complications from two-component columns.
• A functional equation is formulated involving both the first and second derivatives of the generating function, a non-standard feature that increases complexity.
• The functional equation is solved to obtain a generating function for simplex-duplex polyominoes ending with a one-component column.
• The full generating function G(q,1) is then reconstructed from the solution, enabling computation of Taylor series coefficients.
• High-degree Taylor polynomials (up to degree 320) are computed numerically to analyze asymptotic behavior.
• Asymptotic analysis is performed by examining the ratio of successive coefficients and normalizing by the growth constant.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of n-celled simplex-duplex polyominoes, and how does it compare to that of column-convex polyominoes?
• RQ2 Can the area generating function for simplex-duplex polyominoes be derived despite the model's complexity and the absence of a simple recursive structure?
• RQ3 How does the inclusion of two-component columns, under the adjacency restriction, affect the generating function's structure and singularity properties?
• RQ4 To what extent does the use of second derivatives in the functional equation influence the solvability and complexity of the generating function?
• RQ5 Is the resulting generating function for simplex-duplex polyominoes more complex than that of column-convex polyominoes, and what does this imply for future generalizations?

Key findings

• The number of n-celled simplex-duplex polyominoes asymptotically behaves as 0.119442870405 × 3.522019812882^n, based on analysis of a 320-degree Taylor polynomial of G(q,1).
• The dominant singularity of the area generating function G(q,1) is a simple pole located at approximately 0.283927988236.
• The area generating function for column-convex polyominoes is recovered as a special case (w=0) of the bivariate generating function, confirming consistency with known results.
• The functional equation used in the derivation involves both the first and second derivatives of the generating function, a rare and nontrivial feature that increases the formula's complexity.
• The growth constant 3.522020 is significantly higher than that of column-convex polyominoes (3.205569), indicating a denser set of polyominoes.
• Despite the complexity of the generating function, high-precision numerical computation of Taylor coefficients confirms the asymptotic behavior with high confidence.