Overpartitions and Kaur, Rana, and Eyyunni's mex sequences
Brian Hopkins, James A. Sellers
TL;DR
This work addresses the problem of giving bijective proofs that connect the mex sequence statistic for partitions to a family of overpartitions. It introduces a bijection between $P_r^{\mathrm{mex}}(n)$ and the restricted overpartition family $\overline{P}_r(n)$, and proves the key identity $\overline{p}_r(n)=p_r^{\mathrm{mex}}(n)$ through both generating-function and direct combinatorial arguments. The parity-dependent corollaries are established by specialized bijections to $p_e^{>r}(n)$ for odd $r$ and to $p_{o,2}^{>r}(n)$ for even $r$, thereby providing a unified combinatorial interpretation of the previously analytic results. The findings hint at broader connections between mex-based partition statistics and overpartition theory, suggesting avenues for further combinatorial explorations and bijective proofs.
Abstract
Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex sequences and a uniform family of overpartitions which allows us to provide combinatorial proofs of their results, as they requested.