Arithmetic properties of certain partition functions and modular forms

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This thesis studies arithmetic properties of `-regular overpartitions, Andrews' singular overpartitions, overpartitions into odd parts, cubic and overcubic partition pairs, and Andrews' integer partitions with even parts below odd parts. We use various dissections of Ramanujan's theta functions to nd in nite families of arithmetic identities and Ramanujan-type congruences for `-regular overpartitions and overpartitions into odd parts. We nd certain congruences satis ed by A`(n) for ` = 4; 8 and 9, where A`(n) denotes the number of `-regular overpartitions of n. We nd several in nite families of congruences including some Ramanujan-type congruences satis ed by A2`(n) and A4`(n) for any ` 1. We next prove several congruences for po(n) modulo 8 and 16, where po(n) denotes the number of overpartitions of n into odd parts. We also obtain the generating functions for po(16n+2), po(16n+6), and po(16n + 10); and some new p-dissection formulas.
Supervisor: Rupam Barman