A Lower Bound on the Growth Constant of Polyaboloes on the Tetrakis Lattice

Abstract

A “lattice animal” is an edge-connected set of cells on a lattice. In this paper we consider the Tetrakis lattice, and provide the first lower bound on $\lambda _\tau$, the growth constant of polyaboloes (animals on this lattice), proving that $\lambda _\tau \ge 2.4345$. The proof of the bound is based on a concatenation argument and on calculus manipulations. If we also rely on an unproven assumption, which is, however, supported by empirical data, we obtain the conditional slightly-better lower bound 2.4635.

Work on this paper by the first author has been supported in part by ISF Grant 575/15 and by BSF Grant 2017684. Work by the second author has been supported in part by Grant DST-IFA-14-ENG-75 and new faculty Seed Grant NPN5R.

Appendix: Computing Elements of \((\tau (n))\)

We have implemented Redelmeier’s algorithm [14] for counting polyominoes, and adapted it to the Tetrakis lattice.¹ The algorithm was implemented in C on a server with four 2.20 GHz Intel Xeon processors and 512 GB of RAM. The software consisted of about 200 lines of code.

Table 2. Split of $\tau (36)$ into the 16 subtypes $\tau _{i,j}(36)$, $1 \le i,j \le 4$

Fig. 5.

Cell IDs for Redelmeier’s Algorithm

Assume, for simplicity, that we wanted to count polyaboloes up to size n, where n is divisible by 4. Then, we created the graph, dual of the portion of the Tetrakis lattice, that consists of n / 2 columns, each of height $2n+4$. Cells (triangles) of this portion of our lattice were numbered as is shown in Fig. 5(a). In fact, the cell-adjacency graph was identical to the one shown in Fig. 5(b), where a thick edge means that the two cells sharing this edge were not considered as neighbors. In order to count polyaboloes of Types 1, 2, 3, or 4, we fixed their smallest triangle at cell number $n+1$, $n+2$, $n+3$, or n, respectively. This ensured that animals of size n would never spill over the allocated portion of the Tetrakis lattice.

Table 3. Counts of polyaboloes (values of $\tau (21)$–$\tau (36)$ (in bold) are new)

In fact, we needed to count only polyaboloes of two out of the four types, as is implied by Lemma 1. Thus, we ran our program for counting polyaboloes of Types 1 and 2, and computed counts of polyaboloes of Types 3 and 4 by applying the lemma. Then, we summed up the results to finally obtain $\tau (n) = \sum _{i=1}^4 \tau _i(n)$. The running times of our program were 27.5 and 26.25 days, for computing $\tau _1(n)$ and $\tau _2(n)$ for $1 \le n \le 36$, respectively, for a total of 53.75 days for computing $\tau (n)$ for this range of n.

Table 2 provides the split of $\tau (36)$ into all 16 subtypes. Table 3 shows the total counts of polyaboloes, produced by our program, extending significantly the previously-published counts [1, Sequence A197467]. Figure 6 plots the known values of $\root n \of {\tau (n)}$ and $\tau (n)/\tau (n-1)$ for $2 \le n \le 36$, demonstrating the convergence of the two sequences. Figure 7 plots the 36 known values of the sequences $(x_i(n))$ ($i = 1,\dots ,4$), showing the tendencies of these sequences.

Fig. 6.

Convergence of $\root n \of {\tau (n)}$ and $\tau (n)/\tau (n-1)$

Fig. 7.

Empirical monotonicity and convergence of $(x_i(n))$ (for $1 \le i \le 4$)