We introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first give the generating function and establish the $5$-dissections of $k$-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo $5$. Finally, we pose some problems for future work.
"On $k$-restricted overpartitions." Rocky Mountain J. Math. 49 (4) 1207 - 1221, 2019. https://doi.org/10.1216/RMJ-2019-49-4-1207