Dimer-monomer model on the generalized Tower of Hanoi graph

TL;DR

This paper investigates the combinatorial enumeration of dimer-monomers on generalized Tower of Hanoi graphs for dimensions 3 and 4, providing precise entropy bounds and numerical evaluations, and predicting bounds for higher dimensions.

Contribution

It derives tight bounds and high-precision numerical values for the entropy per site of dimer-monomers on specific Hanoi graphs, and extends these results to general dimensions.

Findings

Established bounds for entropy per site with rapid convergence.

Numerical entropy values computed to over a hundred digits.

Predicted entropy bounds for arbitrary dimensions based on lower-dimensional results.

Abstract

We study the number of dimer-monomers $M_d(n)$ on the Tower of Hanoi graphs $TH_d(n)$ at stage $n$ with dimension $d$ equal to 3 and 4. The entropy per site is defined as $z_{TH_d}=\lim_{v \to \infty} \ln M_d(n)/v$, where $v$ is the number of vertices on $TH_d(n)$. We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical value of $z_{TH_d}$ is evaluated to more than a hundred digits correct. Using the results with $d$ less than or equal to 4, we predict the general form of the lower and upper bounds for $z_{TH_d}$ with arbitrary $d$.

Tables

Table 1. Table 1: The values of f 3 ​ ( n ) subscript 𝑓 3 𝑛 f_{3}(n) , g 3 ​ ( n ) subscript 𝑔 3 𝑛 g_{3}(n) , h 3 ​ ( n ) subscript ℎ 3 𝑛 h_{3}(n) , t 3 ​ ( n ) subscript 𝑡 3 𝑛 t_{3}(n) , s 3 ​ ( n ) subscript 𝑠 3 𝑛 s_{3}(n) , M 3 ​ ( n ) subscript 𝑀 3 𝑛 M_{3}(n) with 0 ≤ n ≤ 2 0 𝑛 2 0\leq n\leq 2 .

$n$	0	1	2
$f_3\left(n\right)$	1	1,010	49,464,202,269,253,193
$g_3\left(n\right)$	0	1,242	62,379,666,478,434,024
$h_3\left(n\right)$	1	1,556	78,668,504,245,191,833
$t_3\left(n\right)$	0	1,983	99,212,077,110,534,768
$s_3\left(n\right)$	3	2,571	125,122,091,640,871,731
$M_3\left(n\right)$	10	25,817	1,292,964,293,737,151,090

Table 2. Table 2: The values of α 3 ​ ( n ) subscript 𝛼 3 𝑛 \alpha_{3}(n) , β 3 ​ ( n ) subscript 𝛽 3 𝑛 \beta_{3}(n) , γ 3 ​ ( n ) subscript 𝛾 3 𝑛 \gamma_{3}(n) , ω 3 ​ ( n ) subscript 𝜔 3 𝑛 \omega_{3}(n) with 1 ≤ n ≤ 4 1 𝑛 4 1\leq n\leq 4 . The last digits are rounded off.

$n$	1	2	3	4
${\alpha }_3\left(n\right)$	0.813204508856683	0.792953939347432	0.792939105706120	0.792939105697681
${\beta }_3\left(n\right)$	0.798200514138817	0.792943339611629	0.792939105700090	0.792939105697681
${\gamma }_3\left(n\right)$	0.784669692385275	0.792932741016451	0.792939105694060	0.792939105697681
${\omega }_3\left(n\right)$	0.771295215869312	0.792922143559552	0.792939105688030	0.792939105697681

Table 3. Table 3: The values of ϵ 3 ​ ( n + 1 ) / ϵ 3 ​ ( n ) 2 subscript italic-ϵ 3 𝑛 1 subscript italic-ϵ 3 superscript 𝑛 2 \epsilon_{3}(n+1)/\epsilon_{3}(n)^{2} with 1 ≤ n ≤ 4 1 𝑛 4 1\leq n\leq 4 . The last digits are rounded off.

$n$	1	2	3	4
${ϵ}_3\left(n+1\right)/{ϵ}_3{\left(n\right)}^2$	0.18102932094933	0.17893865402990	0.17893332747848	0.17893332747295

Table 4. Table 4: The values of M 4 ​ ( n ) subscript 𝑀 4 𝑛 M_{4}(n) , f 4 ​ ( n ) subscript 𝑓 4 𝑛 f_{4}(n) , g 4 ​ ( n ) subscript 𝑔 4 𝑛 g_{4}(n) , h 4 ​ ( n ) subscript ℎ 4 𝑛 h_{4}(n) , t 4 ​ ( n ) subscript 𝑡 4 𝑛 t_{4}(n) , s 4 ​ ( n ) subscript 𝑠 4 𝑛 s_{4}(n) and u 4 ​ ( n ) subscript 𝑢 4 𝑛 u_{4}(n) with 0 ≤ n ≤ 2 0 𝑛 2 0\leq n\leq 2 .

$n$	0	1	2
$M_4\left(n\right)$	26	48,645,865	1,209,689,823,065,753,613,801,849,265,389,348,210,254
$f_4\left(n\right)$	1	510,980	12,567,379,442,065,248,794,102,222,711,306,394,841
$g_4\left(n\right)$	0	755,968	18,760,454,431,707,651,977,688,401,100,886,141,664
$h_4\left(n\right)$	1	1,123,642	28,005,432,734,266,093,414,497,192,140,551,929,071
$t_4\left(n\right)$	0	1,677,248	41,806,280,366,033,934,562,540,832,493,986,021,752
$s_4\left(n\right)$	3	2,513,329	62,408,116,726,493,840,561,375,438,310,621,519,011
$u_4\left(n\right)$	0	3,779,500	93,162,456,829,680,622,542,047,599,275,124,003,808

Table 5. Table 5: The values of α 4 ​ ( n ) subscript 𝛼 4 𝑛 \alpha_{4}(n) , ω 4 ​ ( n ) subscript 𝜔 4 𝑛 \omega_{4}(n) , and other ratios with 1 ≤ n ≤ 4 1 𝑛 4 1\leq n\leq 4 . The last digits are rounded off.

$n$	1	2	3	4
${\alpha }_4\left(n\right)$	0.67592808161192	0.66988672837395	0.66988575004178	0.66988575004175
$g_4\left(n\right)/h_4\left(n\right)$	0.67278368021131	0.66988625420357	0.66988575004176	0.66988575004175
$h_4\left(n\right)/t_4\left(n\right)$	0.66993193612394	0.66988578005661	0.66988575004175	0.66988575004175
$t_4\left(n\right)/s_4\left(n\right)$	0.66734120363868	0.66988530593307	0.66988575004173	0.66988575004175
${\omega }_4\left(n\right)$	0.66498981346739	0.66988483183294	0.66988575004172	0.66988575004175

Table 6. Table 6: The entropy per site for dimer-monomers on the Sierpinski gasket S ​ G d 𝑆 subscript 𝐺 𝑑 SG_{d} and Tower of Hanoi graph T ​ H d 𝑇 subscript 𝐻 𝑑 TH_{d} with d = 2 , 3 , 4 𝑑 2 3 4 d=2,3,4 .

$d$	$z_{S{G}_d}$	$z_{T{H}_d}$
2	0.6562942369…	0.5764643016…
3	0.7811514674…	0.6571992114…
4	0.8767794029…	0.7229138308…

Equations

$$z_G=\underset{v(G)\to \infty }{\lim }\frac{\ln N_{DM}(G)}{v(G)},$$

$$e(T{H}_d(n))=\left(\genfrac{}{}{0ex}{}{d+1}{2}\right)\frac{[(d+1{)}^{n+1}-1]}{d}=\frac{(d+1)}{2}[(d+1{)}^{n+1}-1],$$

$$v(T{H}_d(n))=(d+1{)}^{n+1}.$$

$$M_3(n)=f_3(n)+4g_3(n)+6h_3(n)+4t_3(n)+s_3(n)$$

$$P_3(n)$$

$$Q_3(n)$$

$$R_3(n)$$

$$X_3(n)$$

$$Y_3(n)$$

$$W_3(n)$$

$$f_3(n+1)$$

$$g_3(n+1)$$

$$h_3(n+1)$$

$$t_3(n+1)$$

$$s_3(n+1)$$

$$M_3(n)$$

$$f_3(n+1)$$

$$g_3(n+1)$$

$$h_3(n+1)$$

$$t_3(n+1)$$

$$s_3(n+1)$$

$$M_3(n+1)$$

$${\alpha }_3(n)=\frac{f_3(n)}{g_3(n)},{\beta }_3(n)=\frac{g_3(n)}{h_3(n)},{\gamma }_3(n)=\frac{h_3(n)}{t_3(n)},{\omega }_3(n)=\frac{t_3(n)}{s_3(n)},$$

$$0\le {\omega }_3(n)\le {\gamma }_3(n)\le {\beta }_3(n)\le {\alpha }_3(n)\le 1,$$

$$\underset{n\to \infty }{\lim }{\alpha }_3(n)=\underset{n\to \infty }{\lim }{\beta }_3(n)=\underset{n\to \infty }{\lim }{\gamma }_3(n)=\underset{n\to \infty }{\lim }{\omega }_3(n).$$

$${\alpha }_3(n+1)={\omega }_3(n)\frac{A_3(n)}{B_3(n)},{\beta }_3(n+1)={\omega }_3(n)\frac{B_3(n)}{C_3(n)},$$

$${\gamma }_3(n+1)={\omega }_3(n)\frac{C_3(n)}{D_3(n)},{\omega }_3(n+1)={\omega }_3(n)\frac{D_3(n)}{E_3(n)},$$

$$A_3(n)$$

$$B_3(n)$$

$$C_3(n)$$

$$D_3(n)$$

$$E_3(n)$$

$${\alpha }_3(n+1)-{\beta }_3(n+1)={\omega }_3(n)\frac{A_3(n)}{B_3(n)}-{\omega }_3(n)\frac{B_3(n)}{C_3(n)}={\omega }_3(n)\frac{[A_3(n)C_3(n)-B_3(n{)}^2]}{B_3(n)C_3(n)}.$$

$$a_3(n)={\alpha }_3(n)-{\beta }_3(n),b_3(n)={\beta }_3(n)-{\gamma }_3(n),c_3(n)={\gamma }_3(n)-{\omega }_3(n),$$

$$A_3(n)C_3(n)-B_3(n{)}^2$$

Full text

Dimer-monomer model; Tower of Hanoi graph; recursion relations; entropy per site

I Introduction

The dimer-monomer model is an interesting but elusive model in statistical mechanics gaunt69 ; Heilmann70 ; Heilmann72 . In this model, a dimer is realized by a diatomic molecule with two neighboring sites attaching to a surface or lattice. For the sites that are not occupied by any dimers, they could be regarded as covered by monomers. Let us define $N_{DM}(G)$ to be the number of dimer-monomers on a graph $G$.

The computation of the general dimer-monomer model remains to be a difficult problem jerrum , in contrast to the closed-packed dimer problem on planar lattices that had been discussed and solved more than fifty years ago kasteleyn61 ; temperley61 ; fisher61 . Recent computation of close-packed dimers, dimers with a single monomer, and general dimer-monomer models on regular lattices are given in Refs. lu99 ; tzeng03 ; izmailian03 ; izmailian05 ; yan05 ; yan06 ; kong06 ; izmailian06 ; wu06 ; kong06n ; kong06nn . It is also interesting to discuss the dimer-monomer problem on fractals with scaling invariance but not translational invariance. The fractals with non-integer Hausdorff dimension can be constructed from certain basic shape mandelbrot ; Falconer . A famous fractal is the Tower of Hanoi graph, and it has been discussed in different contexts Angeli ; distribution ; Independent .

The dimer-monomer problem on the Tower of Hanoi graph with dimension $d=2$ was discussed in Dimer-monomer . In this article, we shall first recall some basic definitions in section II. In section III, we present the recursion relations for the number of dimer-monomers on $TH_{d}(n)$ with dimension $d=3$, then enumerate the entropy per site using lower and upper bounds in details. The calculation for $TH_{d}(n)$ with dimension $d=4$ will be given in section IV. In the last section, we shall predict the general form of the lower and upper bounds of the entropy per site for dimer-monomers on the Tower of Hanoi graph with arbitrary dimension.

II Preliminaries

In this section, let us review some basic terminology. A graph $G=(V,E)$ that is connected and has no loops is defined by the vertex (site) set $V$ and edge (bond) set $E$ bbook ; fh . Denote $v(G)=|V|$ as the number of vertices in $G$ and $e(G)=|E|$ as the number of edges. Two vertices $a$ and $b$ are neighboring if the edge $ab$ is included in $E$. A matching of a graph $G$ is an independent edge subset where the edges have no common vertices. The number of matching in $G$ is denoted as $N_{DM}(G)$, which corresponds to the number of dimer-monomers in statistical mechanics. Although monomer and dimer weights can be associated to each monomer and dimer (cf. wu06 ), we shall set such weights to 1 here.

$N_{DM}(G)$ can increase exponentially as the number of vertices approaches to infinity, and the entropy per site $z_{G}$ is defined as

[TABLE]

where the subscript $G$ indicates the thermodynamic limit.

The two-dimensional Tower of Hanoi graph $TH_{2}(n)$ at stage $n=0,1,2$ shown in Fig. 1 has been discussed in Ref. Dimer-monomer . At stage $n=0$, $TH_{2}(0)$ is a regular triangle. $TH_{2}(n+1)$ is consisted of three $TH_{2}(n)$ using three edges to connect the outmost vertices. Such arrangement can be generalized to construct the Tower of Hanoi graph with higher dimension. For the general Tower of Hanoi graph $TH_{d}(n)$, the number of edges is

[TABLE]

while the number of vertices is

[TABLE]

The $(d+1)$ outmost vertices of $TH_{d}(n)$ have degree $d$, while the other vertices have degree $(d+1)$. Therefore, in the thermodynamic limit $TH_{d}$ is $(d+1)$-regular.

III The entropy per site for dimer-monomers on $TH_{3}(n)$

We shall consider the entropy per site for dimer-monomers on the three-dimensional Tower of Hanoi graph $TH_{3}(n)$ in details. The following quantities will be used in this section.

Definition III.1

Consider the three-dimensional Tower of Hanoi graph $TH_{3}(n)$ at stage $n$. (i) Define $M_{3}(n)\equiv N_{DM}(TH_{3}(n))$ to be the number of dimer-monomers. (ii) Define $f_{3}(n)$ to be the number of dimer-monomers so that all four outmost vertices are covered by monomers. (iii) Define $g_{3}(n)$ to be the number of dimer-monomers so that one certain outmost vertex (e.g. the topmost vertex shown in Fig. 2) is covered by a dimer and the other three outmost vertices are covered by monomers. (iv) Define $h_{3}(n)$ to be the number of dimer-monomers so that two certain outmost vertices (e.g. the downmost vertices shown in Fig. 2) are covered by monomers and the other two outmost vertices are covered by dimers. (v) Define $t_{3}(n)$ to be the number of dimer-monomers so that one certain outmost vertex (e.g. the topmost vertex shown in Fig. 2) is covered by a monomer and the other three outmost vertices are covered by dimers. (vi) Define $s_{3}(n)$ to be the number of dimer-monomers so that all four outmost vertices are covered by dimers.

These quantities $M_{3}(n)$, $f_{3}(n)$, $g_{3}(n)$, $h_{3}(n)$, $t_{3}(n)$, and $s_{3}(n)$ are illustrated in Fig. 2, where we only show the outmost vertices explicitly. Due to rotational symmetry, there are four orientations of $g_{3}(n)$, six orientations of $h_{3}(n)$ and four orientations of $t_{3}(n)$, so that

[TABLE]

for a non-negative integer $n$. The values of these quantities at $n=0$ are $f_{3}(0)=1$, $g_{3}(0)=0$, $h_{3}(0)=1$, $t_{3}(0)=0$, and $s_{3}(n)=3$, so that $M_{3}(0)=10$. The aim of this section is devoted to the asymptotic behavior of $M_{3}(n)$. The six quantities $M_{3}(n)$, $f_{3}(n)$, $g_{3}(n)$, $h_{3}(n)$, $t_{3}(n)$, $s_{3}(n)$ satisfy the recursion relations given in the following Lemma, and they shall be written as $M$, $f$, $g$, $h$, $t$, and $s$ for simplicity in this section.

We shall define six additional quantities $P_{3}(n)$, $Q_{3}(n)$, $R_{3}(n)$, $X_{3}(n)$, $Y_{3}(n)$, and $W_{3}(n)$ as follows. Let $P_{3}(n)$ be the number of dimer-monomers on $TH_{3}(n)$ so that one certain outmost vertex is covered by a monomer, and the other three outmost vertices can be covered by either dimers or monomers. The other quantities $Q_{3}(n)$, $R_{3}(n)$, $X_{3}(n)$, $Y_{3}(n)$, and $W_{3}(n)$ can be defined similarly as shown in Fig. 3, where no open circle or solid circle is put on the outmost vertex if it can be covered by either a dimer or a monomer. It is clear that

[TABLE]

Lemma III.1

For any non-negative integer $n$, the recursion relations are given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof  The Tower of Hanoi graph $TH_{3}(n+1)$ is composed of four $TH_{3}(n)$ with six additional connecting edges. When a certain connecting edge is contained in the matching, i.e. occupied by a dimer, of $TH_{3}(n+1)$, its vertices in the original $TH_{3}(n)$ must be occupied by monomers. Let us categorized the number of dimer-monomers on $TH_{3}(n+1)$ in terms of the number of six additional edges contained in the matching.

The number $f_{3}(n+1)$ consists of the following cases. (i) ${6\choose 0}=1$ configuration where none of the connecting edges are included in the matching, such that all four constituting $TH_{3}(n)$ are in the $P_{3}(n)$ status. (ii) ${6\choose 1}=6$ configurations where one of the connecting edges is included in the matching, such that two $TH_{3}(n)$ are in the $P_{3}(n)$ status and the other two in the $Q_{3}(n)$ status. (iii) ${6\choose 2}=15$ configurations where two of the connecting edges are included in the matching. Among these configurations, there are 12 of them such that one $TH_{3}(n)$ is in the $P_{3}(n)$ status, two in the $Q_{3}(n)$ status, and one in the $R_{3}(n)$ status. For the other 3 configurations, all four $TH_{3}(n)$ are in the $Q_{3}(n)$ status. (iv) ${6\choose 3}=20$ configurations where three of the connecting edges are included in the matching. Among these configurations, there are 4 of them such that one $TH_{3}(n)$ is in the $P_{3}(n)$ status and the other three in the $R_{3}(n)$ status. There are 4 configurations such that one $TH_{3}(n)$ is in the $f_{3}(n)$ status and the other three in the $Q_{3}(n)$ status. For the other 12 configurations, two $TH_{3}(n)$ are in the $Q_{3}(n)$ status and the other two in the $R_{3}(n)$ status. (v) ${6\choose 4}=15$ configurations where four of the connecting edges are included in the matching. Among these configurations, there are 12 of them such that one $TH_{3}(n)$ is in the $f_{3}(n)$ status, one in the $Q_{3}(n)$ status, and two in the $R_{3}(n)$ status. For the other 3 configurations, all four $TH_{3}(n)$ are in the $R_{3}(n)$ status. (vi) ${6\choose 5}=6$ configurations where five of the connecting edges are included in the matching, such that two $TH_{3}(n)$ are in the $f_{3}(n)$ status and the other two in the $R_{3}(n)$ status. (vii) ${6\choose 6}=1$ configuration where all the connecting edges are included in the matching, such that all four constituting $TH_{3}(n)$ are in the $f_{3}(n)$ status. These configuration are shown in Fig. 4, so that $f_{3}(n+1)$ can be written as

[TABLE]

Here we use the shorthand notations $M$, $P$, $Q$, $R$, $X$, $Y$, $W$ for $M_{3}(n)$, $P_{3}(n)$, $Q_{3}(n)$, $R_{3}(n)$, $X_{3}(n)$, $Y_{3}(n)$, $W_{3}(n)$ in this proof. Using the relations in Eq. (10) for the quantities $P_{3}(n)$, $Q_{3}(n)$, $R_{3}(n)$, Eq. (55) becomes Eq. (15).

By the same token, the recursion relations of $g_{3}(n+1)$, $h_{3}(n+1)$, $t_{3}(n+1)$, $s_{3}(n+1)$ can be expressed as follows.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using the relations in Eq. (10) again, Eqs. (22)-(42) are proved.

Finally, the number of dimer-monomer, $M_{3}(n+1)$, is given by

[TABLE]

as illustrated in Fig. 5. Using the relations in Eq. (10), Eq. (69) becomes Eq. (53), and the proof is completed.  $\Box$

$f_{3}(n)$, $g_{3}(n)$, $h_{3}(n)$, $t_{3}(n)$, $s_{3}(n)$, and $M_{3}(n)$ can be evaluated using Eqs. (15)-(53), and their values for $n=0,1,2$ are listed in Table 1. However, these numbers increase exponentially as $n$ increase, and have no simple integer factorizations.

In the rest part of this section, we shall estimate the entropy per site $z_{TH_{3}}=\lim_{n\to\infty}\ln M_{3}(n)/v(TH_{3}(n))$. For the three-dimensional Tower of Hanoi graph, let us define the ratios

[TABLE]

and their values for $1\leq n\leq 4$ are listed in Table 2. From the first few values of $f_{3}(n)$, $g_{3}(n)$, $h_{3}(n)$, $t_{3}(n)$, $s_{3}(n)$, one can see that $f_{3}(n)<g_{3}(n)<h_{3}(n)<t_{3}(n)<s_{3}(n)$ when $n=1,2$ and it is easy to prove this inequality by induction for all positive integer $n$. Therefore, we have $\alpha_{3}(n)$, $\beta_{3}(n)$, $\gamma_{3}(n)$, $\omega_{3}(n)$ $\in(0,1)$. The relationship of these ratios is given in the following Lemma.

Lemma III.2

For any integer $n>0$, the values of the ratios $\alpha_{3}(n)$, $\beta_{3}(n)$, $\gamma_{3}(n)$, and $\omega_{3}(n)$ are ordered as

[TABLE]

and they are equal in the large $n$ limit

[TABLE]

Proof  It is obvious that all these ratios are positive because $f_{3}(n)$, $g_{3}(n)$, $h_{3}(n)$, $t_{3}(n)$, $s_{3}(n)$ are positive. Let us write $\alpha_{3}(n+1)$, $\beta_{3}(n+1)$, $\gamma_{3}(n+1)$ and $\omega_{3}(n+1)$ as

[TABLE]

where

[TABLE]

Here we use the shorthand notations $\alpha$, $\beta$, $\gamma$, $\omega$ for $\alpha_{3}(n)$, $\beta_{3}(n)$, $\gamma_{3}(n)$, $\omega_{3}(n)$ throughout this section.

We shall demonstrate that $2/3<\omega_{3}(n)<\gamma_{3}(n)<\beta_{3}(n)<\alpha_{3}(n)<1$ by induction on $n$, and prove that the differences between these ratios approach to zero as $n$ increases. The inequality $2/3<\omega_{3}(n)<\gamma_{3}(n)<\beta_{3}(n)<\alpha_{3}(n)<1$ holds for $1\leq n\leq 3$ as shown in Table 2. Let us assume that it remains valid for a certain positive integer $n$.

It is not hard to see that $\alpha_{3}(n)$ decreases while $\omega_{3}(n)$ increases as $n$ increases, namely, $\alpha_{3}(n+1)<\alpha_{3}(n)$ and $\omega_{3}(n)<\omega_{3}(n+1)$, that is shown in the appendix. Define $\epsilon_{3}(n)=\alpha_{3}(n)-\omega_{3}(n)$, and $0\leq\epsilon_{3}(n)\leq 1/10$ holds for $1\leq n\leq 4$. Assume that $\epsilon_{3}(n)>\alpha_{3}(n)-\beta_{3}(n)$, $\epsilon_{3}(n)>\beta_{3}(n)-\gamma_{3}(n)$ and $\epsilon_{3}(n)>\gamma_{3}(n)-\omega_{3}(n)$ for a positive integer $n$, then

[TABLE]

Define the positive quantities

[TABLE]

and write $A_{3}(n)C_{3}(n)-B_{3}(n)^{2}$ in terms of $a_{3}(n)$, $b_{3}(n)$, $c_{3}(n)$, $\omega_{3}(n)$ in descending power of $\omega_{3}(n)$,

[TABLE]

where the summation of the powers of $a_{3}(n)$, $b_{3}(n)$, $c_{3}(n)$ for each term is at least two, and we use the fact that $\epsilon_{3}(n)$ is larger than $a_{3}(n)$, $b_{3}(n)$, or $c_{3}(n)$. It follows that

[TABLE]

because $B_{3}(n)C_{3}(n)>\omega_{3}(n)\sum_{l=0}^{20}\sum_{i,j,k}\omega_{3}(n)^{l}a_{3}(n)^{i}b_{3}(n)^{j}c_{3}(n)^{k}$.

By the same method, we also have $0<\beta_{3}(n+1)-\gamma_{3}(n+1)<\epsilon_{3}(n)^{2}$, $0<\gamma_{3}(n+1)-\omega_{3}(n+1)<\epsilon_{3}(n)^{2}$, such that

[TABLE]

Therefore, we have

[TABLE]

for a positive integer $k\leq n$. Because $0\leq\epsilon_{3}(k)\leq 1/10$ is valid for a small positive integer $k$, $\epsilon_{3}(n)$ decreases as $n$ increases and the values $\alpha_{3}(n)$, $\beta_{3}(n)$, $\gamma_{3}(n)$, $\omega_{3}(n)$ are closed to each other when $n$ becomes large. The value of $\epsilon_{3}(n+1)$ is actually smaller than $\epsilon_{3}(n)^{2}$, and their ratios for $1\leq n\leq 4$ are listed in Table 3. Numerically, we find that in the large $n$ limit

[TABLE]

and the proof is completed.  $\Box$

The lower and upper bounds for dimer-monomers on the three-dimensional Tower of Hanoi graph $TH_{3}(n)$, and the bounds for the entropy per site $z_{TH_{3}}$ are given in the following lemmas.

Lemma III.3

For any positive integer $k\leq n$, the number of dimer-monomer $M_{3}(n)$ is bounded:

[TABLE]

Proof  By Eq. (42) and the simple denotation, we get the upper bound of $s_{3}(n+1)$ as follows.

[TABLE]

where we use the fact that $\alpha_{3}(n)$ is a monotonically decreasing function shown in the appendix, such that

[TABLE]

The lower bound of $M_{3}(n)$ can be obtained similarly.  $\Box$

Lemma III.4

The entropy per site for dimer-monomers on the Tower of Hanoi graph, $z_{TH_{3}}$, is bounded:

[TABLE]

where $k$ is a positive integer and $z_{TH_{3}}=\lim_{n\rightarrow\infty}\frac{\ln{M_{3}(n)}}{v(TH_{3}(n))}$.

Proof  Using Lemma III.3 and $v(TH_{3}(n))=4^{n+1}$ in Eq. (3), the lower and upper bounds for $z_{TH_{3}}$ can be easily derived as follows.

[TABLE]

where the last two terms on the right-hand-side of the inequality approach to zero in the infinite $n$ limit. Similarly,

[TABLE]

and the proof is completed.  $\Box$

Proposition III.1

The entropy per site for dimer-monomers on the Tower of Hanoi graph with $d=3$ in the thermodynamic limit is

[TABLE]

The convergence of the lower and upper bounds is rapid. The value of $z_{TH_{3}}$ is accurate to one hundred and one decimals when $k$ in Lemma III.4 is equal to six.

IV The entropy per site for dimer-monomers on $TH_{4}(n)$

We shall derive the entropy per site for dimer-monomers on the four-dimensional Tower of Hanoi graph $TH_{4}(n)$ in this section. The method is the same as that in the previous section.

Definition IV.1

Consider the four-dimensional Tower of Hanoi graph $TH_{4}(n)$ at stage $n$. (i) Define $M_{4}(n)\equiv N_{DM}(TH_{4}(n))$ to be the number of dimer-monomers. (ii) Define $f_{4}(n)$ to be the number of dimer-monomers so that all five outmost vertices are covered by monomers. (iii) Define $g_{4}(n)$ to be the number of dimer-monomers so that one certain outmost vertex is covered by a dimer while the other four outmost vertices are covered by monomers. (iv) Define $h_{4}(n)$ to be the number of dimer-monomers so that two certain outmost vertices are covered by dimers while the other three outmost vertices are covered by monomers. (v) Define $t_{4}(n)$ to be the number of dimer-monomers so that three certain outmost vertices are covered by dimers while the other two outmost vertices are covered by monomers. (vi) Define $s_{4}(n)$ to be the number of dimer-monomers so that one certain outmost vertex is covered by a monomer while the other four outmost vertices are covered by dimers. (vii) Define $u_{4}(n)$ to be the number of dimer-monomers so that all five outmost vertices are covered by dimers.

$f_{4}(n)$, $g_{4}(n)$, $h_{4}(n)$, $t_{4}(n)$, $s_{4}(n)$ and $u_{4}(n)$ are illustrated in Fig. 6, where we only show the outmost vertices explicitly. Due to rotational symmetry, there are ${5\choose 1}=5$ orientations of $g_{4}(n)$, ${5\choose 2}=10$ orientations of $h_{4}(n)$, ${5\choose 3}=10$ orientations of $t_{4}(n)$, and ${5\choose 4}=5$ orientations of $s_{4}(n)$, such that

[TABLE]

for a non-negative integer $n$. The initial values at $n=0$ are $f_{4}(0)=1$, $g_{4}(0)=0$, $h_{4}(0)=1$, $t_{4}(0)=0$, $s_{4}(n)=3$, $u_{4}(n)=0$, and $M_{4}(0)=26$.

We write a program to get the recursion relations for $TH_{4}(n)$. These recursion relations are much more lengthier than those for $TH_{3}(n)$ and are omitted. The values of $M_{4}(n)$, $f_{4}(n)$, $g_{4}(n)$, $h_{4}(n)$, $t_{4}(n)$, $s_{4}(n)$ and $u_{4}(n)$ for $0\leq n\leq 2$ are listed in Table 4.

For the four-dimensional Tower of Hanoi graph, let us define the ratios $\alpha_{4}(n)=\frac{f_{4}(n)}{g_{4}(n)}$ and $\omega_{4}(n)=\frac{s_{4}(n)}{u_{4}(n)}$, similar to those in Eq. (70). It can be shown that $\alpha_{4}(n)$ decreases monotonically as $n$ increases while $\omega_{4}(n)$ increases monotonically, and

[TABLE]

for any positive integer $n$. The values of these ratios for $1\leq n\leq 4$ are listed in Table 5, and they approach to each other as $n$ increases. In the large $n$ limit, the numerical results give

[TABLE]

By the argument similar to that in Lemmas III.3 and III.4, the entropy per site for dimer-monomers on $TH_{4}(n)$ is bounded:

[TABLE]

where $k$ a positive integer. That leads to the following proposition.

Proposition IV.1

The entropy per site for dimer-monomers on the four-dimensional Tower of Hanoi graph $TH_{4}(n)$ in the thermodynamic limit is $z_{TH_{4}}=0.72291383087181938879...$

The lower and upper bounds given in Eq. (160) converge as rapid as that for the three-dimensional Tower of Hanoi graph $TH_{3}(n)$. The value of $z_{TH_{4}}$ is accurate to one hundred and twenty decimals when $k$ in Eq. (160) is equal to six.

V Summary

The lower and upper bounds of the entropy per site for dimer-monomers on $TH_{2}(n)$ in Ref. Dimer-monomer , and on $TH_{3}(n)$, $TH_{4}(n)$ given above lead to the following conjecture for general $TH_{d}(n)$ with any dimension $d\geq 2$.

Conjecture V.1

Define $\alpha_{d}(n)$ to be the number of dimer-monomers on $TH_{d}(n)$ with all outmost vertices covered by monomers divided by the number with all but one outmost vertices covered by monomers. Define $\omega_{d}(n)$ to be the number of dimer-monomers on $TH_{d}(n)$ with all but one outmost vertices covered by dimers divided by the number with all outmost vertices covered by dimers. Define $\lambda_{d}(n)$ to be the number of dimer-monomers on $TH_{d}(n)$ with all outmost vertices covered by dimers. The entropy per site for dimer-monomers on the $d$-dimensional Tower of Hanoi graph $TH_{d}(n)$ is bounded:

[TABLE]

Notice that although the lower and upper bounds given above are not exactly the same as those in Dimer-monomer , the convergent rate is equivalent. The lower and upper bounds given above apply to $2\leq d\leq 4$, and we conjecture that they are valid for any dimension $d$. It appears that the convergence of the lower and upper bounds of the entropy per site for dimer-monomers on $TH_{d}(n)$ becomes a bit faster as $d$ increases.

The present results can be compared with the entropy per site for dimer-monomers on the Sierpinski gasket (cf. sfs ), $z_{SG_{d}}$, as listed in Table 6. For dimension $d=2,3,4$, the entropy per site on the Tower of Hanoi graph $TH_{d}$ is less than that on the Sierpinski gasket $SG_{d}$, and we conjecture that this relation remains true for arbitrary $d$. This can be attributed to the fact that the degree of $TH_{d}$ is less than that of $SG_{d}$.

Acknowledgements.

This research of S.-C.C. was supported in part by the MOST grant 107-2515-S-006-002.

Appendix A Proof of the monotonicity of $\omega_{3}(n)$ and $\alpha_{3}(n)$

We shall show that $\omega_{3}(n)$ is an ascending function and $\alpha_{3}(n)$ is a descending function here. Using $a,b,c$ to denote $\alpha_{3}(n)-\beta_{3}(n)$, $\beta_{3}(n)-\gamma_{3}(n)$, $\gamma_{3}(n)-\omega_{3}(n)$, respectively, and the definition given in Eq. (75), we find that $\omega_{3}(n+1)$ is always larger than $\omega_{3}(n)$ as follows.

[TABLE]

where the inequality holds since all terms are positive. The relation $\alpha_{3}(n)-\alpha_{3}(n+1)>0$ can be proved similarly, such that $\alpha_{3}(n)$ decreases monotonically as $n$ increases.