Dimer-monomer model on the Sierpinski gasket

We present the numbers of dimer–monomers Md(n) on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four. The upper and lower bounds for the asymptotic growth constant, defined as zSGd=limv→∞lnMd(n)/v where v is the number of vertices on SGd(n), are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of zSGd can be evaluated with more than a hundred significant figures accurate. From the results for d=2,3,4, we conjecture the upper and lower bounds of zSGd for general dimension. The corresponding results on the generalized Sierpinski gasket SGd,b(n) with d=2 and b=3,4 are also obtained.