Refinements of the results on partitions and overpartitions with bounded part differences (Q2279293)

The study on partitions with fixed differences between largest and smallest parts was initiated by \textit{G. E. Andrews} et al. [Proc. Am. Math. Soc. 143, No. 10, 4283--4289 (2015; Zbl 1328.11106)]. \textit{F. Breuer} and \textit{B. Kronholm} [Res. Number Theory 2, Paper No. 2, 15 p. (2016; Zbl 1405.11138)] further studied partitions with bounded difference between largest and smallest parts by a geometric approach. \textit{R. Chapman} [Australas. J. Comb. 64, Part 2, 376--378 (2016; Zbl 1333.05027)] later provided a simplified proof of the Breuer-Kronholm identity. \textit{S. Chern} [Discrete Math. 340, No. 12, 2834--2839 (2017; Zbl 1370.05015)] considered overpartitions with bounded differences between largest and smallest parts. \textit{S. Chern} and \textit{A. J. Yee} [Eur. J. Comb. 70, 317--324 (2018; Zbl 1384.05038)] subsequently gave a refinement of Chern's identity. Soon after, \textit{S. Chern} [N. Z. J. Math. 47, 23--26 (2017; Zbl 1372.05011)] discovered a curious identity. As applications, one can obtain easily the aforementioned results involving overpartitions. Motivated by these works, the author not only gives a bijection for the Breuer-Kronholm identity and obtains a refinement at the same time. Moreover, the author gives a refinement for the Chern-Yee identity. As a consequence, the author also obtains Chern's curious identity [Zbl 1372.05011, loc. cit.] from a combinatorial viewpoint. Finally, the author also gives a refinement of $k$-regular partitions with the difference between largest and smallest part at most $kt$, which were studied by the author in [``k-regular partitions with bounded differences between largest and smallest parts'' (submitted)].