Enumeration of Polyominoes up to Size N=59

TL;DR

This work tackles the problem of enumerating Free($n$) polyominoes by extending counts to $n \le 59$ and verifying prior results up to $n \le 50$. It employs Burnside's Lemma to decompose counts into symmetry classes and uses a transfer-matrix algorithm with Motzkin-path encoding, alongside Redelmeier's rings method for point-symmetric shapes, to compute symmetry-restricted counts. The main contributions are new counts for $51 \le n \le 59$, substantial speedups from multi-threading and symmetry-aware optimizations, and a validated framework for scalable polyomino enumeration. This advances combinatorial enumeration, offering reliable large-scale counts and a methodological blueprint for future work on polyominoes and related discrete geometric structures.

Abstract

This paper reports the results of numerical computations for determining the number of polyominoes of size n (n-ominoes). We verify the existing counts for n <= 50 and newly compute the total number of polyominoes up to n <= 59, extending the counting limit. This work shows that, in addition to optimizing the search algorithm for the polyomino counting problem, multi-threading dramatically improves computational efficiency.

Figures (3)

• Figure 1: Symmetries used (Left) and unused (Right) in this calculation
• Figure 2: Example of M45. The dark gray area is the search domain.
• Figure 3: Example of R180C. Numbers indicate the values of the neighborhood counter.