Graduate Thesis Or Dissertation
 

Periodicity and Partition Congruences

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/ff3659936

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  • 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, Periodicity
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