Dimer coverings on the sierpinski gasket

摘要

We present the number of dimer coverings N (d) (n) on the Sierpinski gasket SG (d) (n) at stage n with dimension d equal to two, three, four or five. When the number of vertices, denoted as v(n), of the Sierpinski gasket is an even number, N (d) (n) is the number of close-packed dimers. When the number of vertices is an odd number, no close-packed configurations are possible and we allow one of the outmost vertices uncovered. The entropy of absorption of diatomic molecules per site, defined as S-sGd =lim(n ->infinity) ln N-d(n)/v(n) is calculated to be ln(2)/3 exactly for SG (2). The numbers of dimers on the generalized Sierpinski gasket SG (d,b) (n) with d=2 and b=3,4,5 are also obtained exactly with entropies equal to ln(6)/7, ln(28)/12, ln(200)/18, respectively. The number of dimer coverings for SG (3) is given by an exact product expression, such that its entropy is given by an exact summation expression. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for SG(d) (n) with d=3,4,5. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of S-SGd with d=3,4,5 can be evaluated with more than a hundred significant figures accurate.