Some conjectures of Ballantine and Merca on truncated sums and the minimal excludant in congruences classes

Abstract

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Since then, a number of results on truncated theta series have been proved. In this paper, we find the connections between truncated sums of certain partition functions and the minimal excludant statistic which has been found to exhibit connections with a handful of objects such as Dyson's crank. We present a uniform method to confirm five conjectures on truncated sums of certain partition functions given by Ballantine and Merca. In particular, we provide partition-theoretic interpretations for some truncated sums by using the minimal excludant in congruences classes.