Two analogues of the crank function are defined for overpartitions -the first residual crank and the second residual crank. This suggests an exploration of crank functions defined for overpartitions whose parts are divisible by an arbitrary d. We examine the positive moments of these crank functions while varying d and prove some inequalities.
In 2009, Corteel, Savelief and Vuletić generalized the concept of overpartitions to a new object called plane overpartitions. In recent work, the author considered a restricted form of plane overpartitions called k-rowed plane overpartions and proved a method to obtain congruences for these and other types of combinatorial generating functions. In this paper, we prove several restricted and unrestricted plane overpartition congruences modulo 4 and 8 using other techniques.2010 Mathematics Subject Classification. 11P83.
Abstract. In this paper, 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, restricted plane overpartition congruences and several examples of restricted partition congruences.
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