A combinatorial proof of a family of truncated identities for the partition function

Abstract

In 2012, Andrews and Merca proved a truncated partition identity by studying the truncated series of Euler's pentagonal number theorem. Andrews and Merca's work has opened up a new study on truncated theta series and a number of results on truncated theta series have been proved in the past decade. Recently, Xia, Yee and Zhao proved a new truncated partition identity by taking different truncated series than the one chosen by Andrews and Merca. Very recently, Yao proved a new truncated identity on Euler's pentagonal number theorem. The identity is equivalent to a family of truncated identities for the partition function which involves the results proved by Andrew-Merca, and Xia-Yee-Zhao. In this paper, we provide a purely combinatorial proof of the family of truncated identities for the partition function. In particular, we answer a question on combinatorial proofs of two partition identities, which were posed by Wang and Xiao.