Periodicity and Partition Congruences

In this note, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our extended technique could be a useful tool to prove congruences for certain types of combinatorial functions based on a bounded number of calculations. As applications of our result, we establish new and existing restricted plane partition congruences and several examples of restricted partition congruences. Also, we define a restricted form of plane overpartitions called k rowed plane overpartitions as plane overpartitions with at most k rows. We derive the generating function for this type of partition and obtain a congruence modulo 4. Next, we engage a combinatorial technique to establish plane and restricted plane overpartition congruences modulo small powers of 2. For each even integer k, we prove a set of k-rowed plane overpartition congruences modulo 4. For odd integer k, we prove an equivalence relation modulo 4 between k-rowed plane overpartitions and unrestricted overpartitions. As a consequence, using a result of Hirschhorn and Sellers, we obtain an infinite family of k rowed plane overpartition congruences modulo 4 for each odd integer k ≥ 1. Also, we obtain a few unrestricted plane overpartition congruences modulo 4. We establish and prove several restricted plane overpartition congruences modulo 8. Some examples of equivalences modulo 4 and 8 between plane overpartitions and overpartitions are obtained. In addition, we find and prove an infinite family of 5-rowed plane overpartition congruences modulo 8.Keywords: Partitions, Plane Overpartitions, Plane Partitions, Periodicit

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.Comment: 35 pages, 13 figures. arXiv admin note: substantial text overlap with arXiv:1007.408