# Power series expansions for the planar monomer-dimer problem

## TL;DR

This paper develops a computational method to generate extensive power series expansions for the free energy of the planar monomer-dimer model, surpassing previous results despite high complexity.

## Contribution

It introduces a new approach to compute longer power series expansions for the monomer-dimer problem, enabling more accurate bounds and numerical estimates.

## Key findings

- Computed nearly three times more terms than previous work.
- Provided tighter bounds and more precise numerical values for free energy.
- Demonstrated potential applicability to other complex models.

## Abstract

We compute the free energy of the planar monomer-dimer model. Unlike the classical planar dimer model, an exact solution is not known in this case. Even the computation of the low-density power series expansion requires heavy and nontrivial computations. Despite of the exponential computational complexity, we compute almost three times more terms than were previously known. Such an expansion provides both lower and upper bound for the free energy, and allows to obtain more accurate numerical values than previously possible. We expect that our methods can be applied to other similar problems.