Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions p −k (n) for all k ≥ 2. This enables us to extend the k-colored partition function multiplicatively to a function on k-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.
Motivated by the recent work of Hirschhorn on vanishing coefficients of the arithmetic progressions in certain q-series expansions, we study some variants of these q-series and prove some comparable results. For instance, leta 1 (n)q n , then a 1 (5n + 3) = 0.
Abstract. A generalized crank (k-crank) for k-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined k-crank. Namely, we first obtain some inequalities between the k-crank counts M k (r, m, n) for m = 2, 3 and 4, then we prove the positivity of symmetrized even k-crank moments weighted by the parity for k = 2 and 3. We conclude with several remarks on furthering the study initiated here.
Abstract. Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our methods are elementary relying entirely on the three classical theta functions, and are motivated by the seminal work of Ramanujan, Garvan, Hammond and Lewis.
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