# 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.

## Key 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 $d$.