Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Abstract

A d-dimensional polycube is a face-connected set of cells on $\mathbb {Z}^d$. Let $A_d(n)$ denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and $\lambda _d$ denote their growth constant $\displaystyle \lim _{n \rightarrow \infty } {\scriptstyle \frac{A_d(n+1)}{A_d(n)}}$. We revisit and extend the method for the best known upper bound on $A_2(n)$. Our contributions include the following.

• 1. We prove that $\lambda _2 \le 4.5252$;

• 2. We prove that $\lambda _d \le (2d-2)e+o(1)$ for $d \ge 2$ (already improving significantly the known upper bound on $\lambda _3$ to 9.8073); and

• 3. We implement an iterative process in three dimensions, improving further the upper bound on $\lambda _3$ to 9.3835.

Appendices

Appendix A: Singularity Analyses of the Diagonals of the Generating Functions in $d=2,3$

In this appendix, we use a standard method for analyzing the singularities of the diagonals of our bivariate functions.

1.1 A.1 $d=2$

Our goal is to analyze the diagonal of the bivariate generating function

$$\begin{aligned} g^{(2)}(x,y) = \frac{x}{1-y\big ((1+x)^2 + x^2\big )}. \end{aligned}$$

By using the standard transformation $G(x,y) := g^{(2)}(x/y,y)/y$, the diagonal coefficients of $g^{(2)}(\cdot ,\cdot )$ form the coefficient of $y^{-1}(\cdot )$ in $G(\cdot ,\cdot )$. Now, by applying residue analysis, and, in particular, the theorem about the residues of a rational function, we find that the diagonal is the function

$$\begin{aligned} q(x)=-\frac{(2x-1) \sqrt{1-4x-4x^2}+(1-4x-4x^2)}{4x(1-4x-4x^2)}. \end{aligned}$$

Hence, we need to determine the asymptotic behavior of $[x^n]q(x)$. To this aim, we use the singularity-analysis method of Flajolet and Sedgewick [11, P. 393, Thm. VI.4]. Our final goal is to find the dominant singularity r of q(x), and to find an asymptotically equivalent function of the form $(1-z)^\alpha$, for which asymptotic expressions are known.

In this case, the dominant singularity is one of the roots of $1{-}4x{-}4x^2$, namely, $r=(\sqrt{2}-1)/2 \approx 0.21$. (The other root is $s=-(\sqrt{2}+1)/2 \approx -1.21$.) We rewrite q(x) as $-\frac{2x-1}{4x}(1-4x-4x^2)^{-1/2} \ -1/(4x)$. The term $-1/(4x)$ is negligible for the asymptotics. For $x=r$, the prefactor $-\frac{2x-1}{4x}$ yields $1/\sqrt{2}$.

In order to separate the singular part of $(1-4x-4x^2)^{-1/2}$, and then to use the standard result concerning the asymptotics of $(1-z)^\alpha$, we use the substitution $z=x/r$ and rewrite

$$\begin{aligned} 1-4x-4x^2 = -4(x-s)(x-r) = -4(rz-s)(rz-r) = 4(rz-s)r(1-z). \end{aligned}$$

For the part $\big (4(rz-s)r\big )^{-1/2}$, we can just substitute $x=r$ (equivalently, $z=1$) because this part is not singular at this point. We have that $4(rz-s)r = 2 \sqrt{2} (\sqrt{2}-1)$, and, thus, we obtain another prefactor $1/\sqrt{2(2-\sqrt{2})}$.

Finally, for the part $(1-z)^{-1/2}$ which is singular at $x=r$, we use the corresponding “standard scale” result, $[z^n](1-z)^{-1/2} \sim \frac{1}{\sqrt{\pi n}}$ [ibid, P. 388]. Therefore, $[x^n](1-x/r)^{-1/2} \sim \frac{1}{\sqrt{\pi n}} (1/r)^n$, and we finally have that

$$\begin{aligned}{}[x^n]q(x) \sim \frac{1}{2\sqrt{2-\sqrt{2}}} \, \frac{1}{\sqrt{\pi n}} \, \Big (2(\sqrt{2}+1)\Big )^n, \end{aligned}$$

concluding that the growth constant of this function is $2(\sqrt{2}+1) \approx 4.82843$, as we already found in Sect. 4.1.

1.2 A.2 $d=3$

The analysis of the 3-dimensional case in much more involved than that of the 2-dimensional case. We describe here only the main ingredients of the analysis without getting into the many technical difficulties. In this case, it is not possible to write the generating function q(x) for the diagonal explicitly, however, we can still determine its asymptotic behavior using symbolic computation. Using Newton polygon in order to determine the number of small roots and resultants, and then finding a polynomial equation satisfied by the sum of (implicitly given) residues, we obtain the equation

$$\begin{aligned} \begin{aligned}&q^4(x) ( 272x^5 + 800x^4 + 708x^3 + 184x^2 - 27x ) \\&\quad + q^2(x) ( -8x^3 - 24x^2 +5x ) + 8q(x)x - q(x) + x = 0. \end{aligned} \end{aligned}$$

In order to determine the dominant singularity, we consider the exceptional set, where the discriminant of this equation vanishes (see [11, Sec. VII.7.1]). This discriminant has two positive roots, approximately 0.1020 and 0.4638.⁶ The growth constant of our sequence is obviously larger than 3. (This can be justified in many ways, for example, by the fact that the growth constant in two dimensions is already greater than 4.) Therefore, we conclude that the dominant singularity is about $r \approx 0.101965$, and, accordingly, the growth constant is $1/r \approx 9.8073$, as we already found in Sect. 4.2.

Appendix B: Klarner and Rivest’s Second Part of Condition ($\varvec{*}$) is Incorrect

In this appendix, we explain why we believe that Klarner and Rivest did not use in their program the second part of condition ($\varvec{*}$) (marked with a red frame), as they state it in their paper [19, p. 600]:

1.1 B.1 Argument 1

Consider the set $\textsf {C} _4$, that contains all twigs that have exactly four black cells, or fewer black cells and no white cells. Let us enumerate the twigs that have one, two, or three black cells and no open cells. We can easily observe the following.

• 1. There is only one twig ($L_1 \in \textsf {L}$, shown in Fig. 4) with one dead cell and no open cells.

• 2. There are only two twigs (see Fig. 10) with two dead cells and no open cells.

• 3. There are only six twigs with three dead cells and no open cells. The sequences corresponding to these six twigs are $L_2 *L_2 *L_1$, $L_2 *L_4 *L_1$, $L_3 *L_1 *L_1$, $L_4 *L_2 *L_1$, $L_4 *L_4 *L_1$, and $L_5 *L_1 *L_1$.

Fig. 10

The only two twigs with two dead cells and no open cells

We now show that there are 400 twigs with four dead cells and no open cells, which, together with twigs with less dead cells, sum up to a total of 409 twigs in the set $\textsf {C} _4$. Each twig T with 4 dead cells corresponds to a sequence $(\alpha , \beta , \gamma , \delta )$ of four elements of $\textsf {L}$ (see Fig. 4), such that $T= \alpha *\beta *\gamma *\delta$. (Note that T contains its forming twigs in reverse order.) Trivially, in total, there are $|\textsf {L} |^4=5^4=625$ sequences of four elements of $\textsf {L}$. However, some of these sequences are invalid, namely, they do not represent valid twigs. For example, the only valid sequence that starts with $L_1$ is of length 1 since $L_1$ has no open cells. It is easy to verify that the following sequences are exactly those that are invalid since their prefixes correspond to configurations with less than four dead cells and no open cells: $S_1=(L_1,\beta ,\gamma ,\delta )$, $S_2=(L_2,L_1,\gamma ,\delta )$, $S_3=(L_4,L_1,\gamma ,\delta )$, $S_4=(L_2,L_4,L_1,\delta )$, $S_5=(L_4,L_2,L_1,\delta )$, $S_6=(L_2,L_2,L_1,\delta )$, $S_7=(L_4,L_4,L_1,\delta )$, $S_8=(L_3,L_1,L_1,\delta )$, and $S_9=(L_5,L_1,L_1,\delta )$.

Clearly, we have that $|S_1|=5^3=125$, $|S_2|=|S_3|=5^2=25$, and $|S_4|=|S_5|=|S_6|=|S_7|=|S_8|=|S_9|=5$. There remain exactly twenty invalid sequences, as we proceed to show. The twigs $L_4$ and $L_5$ (see Fig. 4) cannot be concatenated to any of the configurations (a–j) (Figs. 11 and 12) since this would violate the first part of condition ($\varvec{*}$), that is, a cell of $L_4$ or $L_5$ would overlap with an occupied cell of the configuration or a cell marked as forbidden. This results in $2{\cdot }10=20$ more invalid sequences. Thus, we find that the number of twigs with four dead cells is

$$\begin{aligned} |\textsf {L} |^4 - \sum _{i=1}^{9} |S_i| - 20 = 400. \end{aligned}$$

Hence, adding items (1–3) above (twigs with less than four dead cells), we obtain that $|\textsf {C} _4| = 400 + 1 + 2 + 6 = 409$, which is precisely the number provided for $|\textsf {C} _4|$ by Klarner and Rivest as well (see Table 1). On the other hand, restricting the construction of $\textsf {C} _4$ further with the second part of condition ($\varvec{*}$) would imply that, for example, concatenating $L_1$ to configuration (d) is invalid, which would decrease the size of $\textsf {C} _4$. Therefore, we conclude that Klarner and Rivest did not use the second part of condition ($\varvec{*}$).

Fig. 11

Concatenations starting with $L_3 *L_2$

Fig. 12

Concatenations starting with $L_3 *L_3$

1.2 B.2 Argument 2

Consider P, the pentomino shown in Fig. 13. The spanning tree of P corresponds to the sequence $L_3 {*} L_2 {*} L_4 {*} L_1 {*} L_1$ (which is equivalent to concatenating $L_1$ twice to configuration (d) in Fig. 11). In terms of elements of $\textsf {C} _4$, this is a sequence of length two: its first element T is the twig corresponding to adding $L_1$ to configuration (d), and its second element is $L_1$. By definition, the two twigs T and $L_1$ belong to $\textsf {C} _4$. However, if we were to use the second part of condition ($\varvec{*}$), then T would be discarded as an element of $\textsf {C} _4$ because a forbidden cell of $L_1$ would overlap with a cell of T. In such a situation, $\textsf {C} _4$ would not be a complete set of building blocks for polyominoes, and P would have no corresponding sequence of weight $x^5 y^5$ of elements of $\textsf {C} _4$. Therefore, the method would have failed to provide an upper bound on $\lambda _2$ if the second part of condition ($\varvec{*}$) had been used, as some polyominoes (such as P) would be overlooked.

Fig. 13

The ‘P’ pentomino

Appendix C: Weight Function in Two Dimensions

The following is the weight function $W_{21}(x,y)$, from which we computed the upper bound $\lambda _2 \le 4.5252$.

$$\begin{aligned} W_{21}(x,y)= & {} 1152x^{41}y^{21} +96256x^{40}y^{21} +2056288x^{39}y^{21} +24172920x^{38}y^{21} \\&+187337816x^{37}y^{21} +1070722580x^{36}y^{21} +4785628001x^{35}y^{21}\\&+17443616434x^{34}y^{21} +53301072439x^{33}y^{21} +139441849789x^{32}y^{21} \\&+316994896043x^{31}y^{21} + 633630643451x^{30}y^{21} +1123186767205x^{29}y^{21}\\&+1777867306642x^{28}y^{21} + 2521701005541x^{27}y^{21} +3204911295924x^{26}y^{21} \\&+3628247213386x^{25}y^{21} + 3615763450791x^{24}y^{21} +3089960078272x^{23}y^{21}\\&+2115755445262x^{22}y^{21} + 960344267887x^{21}y^{21} +238239106019x^{20}y^{21} \\&+59145846645x^{19}y^{20} + 14696358415x^{18}y^{19} +3655351652x^{17}y^{18}\\&+910239465x^{16}y^{17} +226975964x^{15}y^{16} +56691193x^{14}y^{15}\\&+14187563x^{13}y^{14} +3559132x^{12}y^{13} +895513x^{11}y^{12} \\&+ 226165x^{10}y^{11} +57394x^9y^{10} +14657x^8y^9 +3775x^7y^8\\&+984x^6y^7 +261x^5y^6 +71x^4y^5 +20x^3y^4 +6x^2y^3 +2xy^2 +y \end{aligned}$$

Appendix D: Weight Function in Three Dimensions

The following is the weight function for $i=9$, from which we computed the upper bound $\lambda _3 \le 9.3835$.

$$\begin{aligned} W_9(x,y)= & {} y +4xy^2 +23x^2y^3 +150x^3y^4 +1051x^4y^5 +7661x^5y^6 \\&+57337x^6y^7 +437050x^7y^8 +3376485x^8y^9 +26352274x^9y^9 \\&+108757201x^{10}y^9 +306714778x^{11}y^9 +674917794x^{12}y^9 \\&+1222175063x^{13}y^9 +1866911075x^{14}y^9 +2434995919x^{15}y^9\\&+2728046412x^{16}y^9 +2631637304x^{17}y^9 +2185885771x^{18}y^9\\&+1560584567x^{19}y^9 +954538066x^{20}y^9 +497886496x^{21}y^9\\&+220105634x^{22}y^9 +81810253x^{23}y^9 +25294655x^{24}y^9\\&+6411687x^{25}y^9 +1305352x^{26}y^9 +207134x^{27}y^9 \\&+24462x^{28}y^9 +1992x^{29}y^9 +97x^{30}y^9 +2x^{31}y^9 \end{aligned}$$

Appendix E: Computing the Radius of Convergence

Let f(x,y) be a rational bivariate generating function. Klarner and Rivest [19, §3] showed how to compute the radius of convergence of the diagonal of f(x,y). This requires a change of variable in order to apply the residue theorem. The diagonal function $f_D(z) = \sum _n l(n,n)z^n$ could then be written as a sum of residues. The following is our Maple implementation of this method. (The function f(x, y) corresponds to the generating function from Sect. 3.1.)