Linked partition ideals and overpartitions

Abstract

Linked partition ideals which were first introduced by Andrews have recently appeared in a series of works to study generating functions for partitions. Recently, Andrews found some relations between a certain kind of overpartitions and 4-regular partitions into distinct parts. Then with the aid of linked partition ideals for overpartitions, Andrews and Chern established a general relation between these two sets of partitions. Motivated by their work, we consider the overpatitions denoted by $A^k$ satisfying the following conditions: (1) Only odd parts may be overlined; (2) The difference between any two parts is $⩾2k$ where the inequality is strict if the larger one is overlined. Let S be a set of given parts. Then $A_S^k$ denotes the subset of overpartitions in $A^k$ where parts from S are forbidden. Combining linked partition ideals and a recurrence relation for a family of multiple series given by Chern, we study the generating functions for $A_S^k$ for some given S. Furthermore, by establishing a q-series identity, we find a relation between $A_{\left\{\bar{1}\right\}}^1$ and distinct partitions. Meanwhile, some statistics on partitions are discussed.