Identities involving the number of missing integers in partitions - combinatorial proofs

A missing integer in a partition $\lambda$ is a positive integer less than the largest part of $\lambda$ that does not appear as a part in $\lambda$. In the recent paper On the number of missing integers in partitions, Bhoria, Eyyunni, and Santra examined the number of partitions (overpartitions) of $n$ with a fixed number of missing integers and established generating functions, identities, and congruences for them. In this article, we provide combinatorial proofs for several of their results.