This paper investigates representations of real numbers with an arbitrary negative base ˇ< 1, which we call the . ˇ/-expansions. They arise from the orbits of the . ˇ/-transformation which is a natural modification of theˇ-transformation. We show some fundamental properties of . ˇ/-expansions, each of which corresponds to a well-known fact of ordinaryˇ-expansions. In particular, we characterize the admissible sequences of . ˇ/-expansions, give a necessary and sufficient condition for the . ˇ/-shift to be sofic, and explicitly determine the invariant measure of the . ˇ/-transformations.

We study a generalization of Holte's amazing matrix, the transition probability matrix of the Markov chains of the 'carries' in a non-standard numeration system. The stationary distributions are explicitly described by the numbers which can be regarded as a generalization of the Eulerian numbers and the MacMahon numbers. We also show that similar properties hold even for the numeration systems with the negative bases.MSC: 60C05, 60J10, 05E99

As a continuation to our previous work [7], we consider a generalization of carries process. Our results are : (i) right eigenvectors of the transition probability matrix, (ii) correlation of carries between different steps, and (iii) generalized riffle shuffle whose corresponding descent process has the same distribution as that of the generalized carries process.

We consider the dimer problem on a non-bipartite graph G, where there are two types of dimers one of which we regard impurities. Results of simulations using Markov chain seem to indicate that impurities are tend to distribute on the boundary, which we set as a conjecture. We first show that there is a bijection between the set of dimer coverings on G and the set of spanning forests on two graphs which are made from G, with configuration of impurities satisfying a pairing condition. This bijection can be regarded as a extension of the Temperley bijection. We consider local move consisting of two operations, and by using the bijection mentioned above, we prove local move connectedness. We further obtained some bound of the number of dimer coverings and the probability finding an impurity at given edge, by extending the argument in [7].Mathematics Subject Classification (2000): 60C05, 82B20.

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