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We derive exactly the number of Hamiltonian paths H(n) on the two dimensional Sierpinski gasket SG(n) at stage $n$, whose asymptotic behavior is given by $frac{sqrt{3}(2sqrt{3})^{3^{n-1}}}{3} times (frac{5^2 times 7^2 times 17^2}{2^{12} times 3^5 times 13})(16)^n$. We also obtain the number of Hamiltonian paths with one end at a certain outmost vertex of SG(n), with asymptotic behavior $frac {sqrt{3}(2sqrt{3})^{3^{n-1}}}{3} times (frac {7 times 17}{2^4 times 3^3})4^n$. The distribution of Hamiltonian paths on SG(n) with one end at a certain outmost vertex and the other end at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean $ell$ displacement between the two end vertices of such Hamiltonian paths on SG(n) is $ell log 2 / log 3$ for $ell>0$.
The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski gasket $SG
We present the number of dimers $N_d(n)$ on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three, four or five, where one of the outmost vertices is not covered when the number of vertices $v(n)$ is an odd number. The e
We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of config
The multifractal behavior of the normalized first passage time is investigated on the two dimensional Sierpinski gasket with both absorbing and reflecting barriers. The normalized first passage time for Sinai model and the logistic model to arrive at
We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket $SG_{2,b}(n)$ at stage $n$ with $b$ equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth