Combinatorial Extensions on Andrews and El Bachraoui's Almost-Distinct Partitions

Andrews and El Bachraoui recently studied integer partitions where the smallest part is repeated a specified number of times and any other parts are distinct. Their results included two ``surprising identities'' for which they requested combinatorial proofs. We provide combinatorial proofs for an infinite family of related identities and also consider the analogous partitions where only the largest part can be repeated. The results give rise to infinite families of inequalities for the number of partitions with distinct parts.