Study of dimer–monomer on the generalized Hanoi graph

Abstract

We study the number of dimer–monomers $M_d(n)$ on the Hanoi graphs $H_d(n)$ at stage n with dimension d equal to 3 and 4. The entropy per site is defined as $z_{H_d}=\lim _{v \rightarrow \infty } \ln M_d(n)/v$, where v is the number of vertices on $H_d(n)$. 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 values of $z_{H_d}$ for $d=3, 4$ are 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 $z_{H_d}$ with arbitrary d.

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 $\alpha _3(n)$ 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. (21), we find that $\omega _3(n+1)$ is always larger than $\omega _3(n)$ as follows.

$$\begin{aligned}&\omega _3(n+1)-\omega _3(n) =\frac{\omega _3(n)}{E_3(n)}(D_3(n)-E_3(n))\nonumber \\&\quad =\frac{\omega _3(n)}{E_3(n)}[(64a+64c+64b)\omega ^{11}\nonumber \\&\qquad +(256ab+768bc+384b+512c^2+384c+256b^2+288a+512ac)\omega ^{10}\nonumber \\&\qquad +(2688c^2+1792c^3+3584bc^2+1152c+2176b^2c+3840bc+2016ac+624a\nonumber \\&\qquad +384b^3+384ab^2+864ab+1792ac^2+1792abc+1104b+1152b^2)\omega ^9\nonumber \\&\qquad +(2352b^2+7680b^2c^2+8064c^3+9120bc+5376abc^2+8960bc^3+256ab^3+256b^4\nonumber \\&\qquad +1968b+2304ab^2c+840a+5184abc+1152b^3+3584ac^3+6912c^2+1392ab\nonumber \\&\qquad +8064b^2c+864ab^2+2232c+6048ac^2+3584c^4+2560cb^3+3744ac+14976bc^2)\omega ^8\nonumber \\&\qquad +(1392b^3+2808b^2+13296b^2c+13200bc+1416ab+288ab^3+9360ac^2+6960abc\nonumber \\&\qquad +4320ab^2c+912ab^2+1280ab^3c+384b^4+6144cb^3+64ab^4+1344b^4c+4224ac\nonumber \\&\qquad +64b^5+3072c+2400b+768a+11184c^2+17280c^3+13440c^4+29040bc^2+30720bc^3\nonumber \\&\qquad +23040b^2c^2+10080ac^3+12960abc^2+4480c^5+13440bc^4+14720b^2c^3+7040b^3c^2\nonumber \\&\qquad +4480ac^4+8960abc^3+5760ab^2c^2)\omega ^7\nonumber \\&\qquad +(864b^3+2148b^2+12432b^2c+256ab^4c+12720bc+1020ab+144ab^3+8520ac^2\nonumber \\&\qquad +5736abc+3648ab^2c+600ab^2+1152ab^3c+144b^4+5760cb^3+1536b^4c+256b^5c\nonumber \\&\qquad +3168ac+3132c+2088b+492a+12432c^2+22440c^3+23040c^4+33216bc^2\nonumber \\&\qquad +47040bc^3+29664b^2c^2+12480ac^3+13920abc^2+3584c^6+13440c^5+12544bc^5\nonumber \\&\qquad +16640b^2c^4+10240b^3c^3+36480bc^4+34560b^2c^3+13056b^3c^2+2816b^4c^2+10080ac^4\nonumber \\&\qquad +17280abc^3+8640ab^2c^2+8960abc^4+7680ab^2c^3+2560ab^3c^2+3584ac^5)\omega ^6\nonumber \\&\qquad +(384ab^4c^2+300b^3+1086b^2+7188b^2c+8436bc+552ab+24ab^3+4992ac^2\nonumber \\&\qquad +3252abc+1872ab^2c+252ab^2+432ab^3c+24b^4+2688cb^3+432b^4c+384b^5c^2\nonumber \\&\qquad +1638ac+2404c+1312b+220a+9792c^2+19008c^3+22560c^4+1792c^7\nonumber \\&\qquad +24192bc^2+7168bc^6+11136b^2c^5+8320b^3c^4+40224bc^3+20592b^2c^2+8640ac^3\nonumber \\&\qquad +8784abc^2+8064c^6+17280c^5+25344bc^5+28800b^2c^4+5376abc^5+5760ab^2c^4\nonumber \\&\qquad +2560ab^3c^3+13824b^3c^3+1792ac^6+2944b^4c^3+41520bc^4+32736b^2c^3+8928b^3c^2\nonumber \\&\qquad +2304b^4c^2+9360ac^4+13920abc^3+5472ab^2c^2+12960abc^4+8640ab^2c^3\nonumber \\&\qquad +1728ab^3c^2+6048ac^5)\omega ^5 \end{aligned}$$

$$\begin{aligned}&\qquad +(256b^5c^3+1792abc^6+2304ab^2c^5+1280ab^3c^4+512ac^7+256ab^4c^3+60b^3\nonumber \\&\qquad +364b^2+2688b^2c+3870bc+226ab+1962ac^2+1326abc+600ab^2c+60ab^2\nonumber \\&\qquad +72ab^3c+744cb^3+72b^4c+590ac+1386c+588b+66a+5462c^2+10584c^3\nonumber \\&\qquad +13152c^4+2688c^7+11460bc^2+9600bc^6+12672b^2c^5+7296b^3c^4+20256bc^3\nonumber \\&\qquad +8364b^2c^2+512c^8+1536b^4c^4+2304bc^7+4096b^2c^6+3584b^3c^5+3648ac^3\nonumber \\&\qquad +3636abc^2+6912c^6+11400c^5+19104bc^5+17904b^2c^4+5184abc^5+4320ab^2c^4\nonumber \\&\qquad +1152ab^3c^3+6144b^3c^3+2016ac^6+1536b^4c^3+23856bc^4+15264b^2c^3+2880b^3c^2\nonumber \\&\qquad +432b^4c^2+4440ac^4+6096abc^3+2016ab^2c^2+6960abc^4+3648ab^2c^3+432ab^3c^2\nonumber \\&\qquad +3744ac^5)\omega ^4\nonumber \\&\qquad +(864abc^6+864ab^2c^5+288ab^3c^4+288ac^7+6b^3+81b^2+670b^2c+1212bc\nonumber \\&\qquad +66ab+526ac^2+388abc+120ab^2c+6ab^2+120cb^3+147ac+588c+180b\nonumber \\&\qquad +12a+2094c^2+3718c^3+4320c^4+1152c^7+3518bc^2+3600bc^6+3888b^2c^5\nonumber \\&\qquad +1584b^3c^4+64ac^8+640b^2c^7+640b^3c^6+320b^4c^5+64b^5c^4+256abc^7+64c^9\nonumber \\&\qquad +320bc^8+5964bc^3+2118b^2c^2+384ab^2c^6+256ab^3c^5+64ab^4c^4+384c^8+384b^4c^4\nonumber \\&\qquad +1536bc^7+2304b^2c^6+1536b^3c^5+978ac^3+996abc^2+2352c^6+3648c^5+5712bc^5\nonumber \\&\qquad +4440b^2c^4+1392abc^5+912ab^2c^4+144ab^3c^3+1152b^3c^3+624ac^6+144b^4c^3\nonumber \\&\qquad +6816bc^4+3756b^2c^3+588b^3c^2+72b^4c^2+1152ac^4+1596abc^3+444ab^2c^2\nonumber \\&\qquad +1704abc^4+816ab^2c^3+72ab^3c^2+960ac^5)\omega ^3\nonumber \\&\qquad +(12b^2+111b^2c+255bc+12ab+96ac^2+81abc+12ab^2c+12cb^3+24ac\nonumber \\&\qquad +175c+34b+a+513c^2+738c^3+666c^4+24c^7+684bc^2+96bc^6+144b^2c^5\nonumber \\&\qquad +96b^3c^4+978bc^3+324b^2c^2+162ac^3+174abc^2+144c^6+396c^5+432bc^5\nonumber \\&\qquad +432b^2c^4+72abc^5+72ab^2c^4+24ab^3c^3+144b^3c^3+24ac^6+24b^4c^3+852bc^4\nonumber \\&\qquad +516b^2c^3+60b^3c^2+162ac^4+222abc^3+60ab^2c^2+192abc^4+96ab^2c^3+96ac^5)\omega ^2\nonumber \\&\qquad +(18b^2c^3+30b^2c^2+15ac^3+12ac^2+18bc^4+12abc+68c^2+33c+36bc+6c^5\nonumber \\&\qquad +b^2+30c^4+3b+75bc^2+12abc^3+2ac+ab+60bc^3+6ab^2c^2+15abc^2+6b^3c^2\nonumber \\&\qquad +63c^3+6ac^4+12b^2c)\omega \nonumber \\&\qquad +2bc^2+ac^2+b^2c+3c+3bc+abc+3c^2+c^3]>0 , \end{aligned}$$

(42)

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.