Dimer coverings on the Tower of Hanoi graph

Abstract

We present the number of dimer coverings Nd(n) on the Tower of Hanoi graph THd(n) at n stage with dimension 2 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. When the number of vertices v(n) is even, Nd(n) gives the number of close-packed dimers; when the number of vertices is odd, it is impossible to have a close-packed configurations and one of the outmost vertices is allowed to be unoccupied. We define the entropy of absorption of diatomic molecules per vertex as S[Formula: see text][Formula: see text]=[Formula: see text][Formula: see text] Nd(n)/v(n), that can be shown exactly for TH2, while its lower and upper bounds can be derived in terms of the results at a certain n for THd(n) with 3 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. We find that the difference between the lower and upper bounds converges rapidly to zero as n increases, such that the value of S[Formula: see text] with d[Formula: see text]=[Formula: see text]3 and 5 can be calculated with at least 100 correct digits.