A Minimal Excludant over Overpartitions

Define the minimal excludant of an overpartition $\pi$ , denoted $\overline{{\mathrm{mex}}}(\pi)$ , to be the smallest positive integer that is not a part of the non-overlined parts of $\pi$ . For a positive integer $n$

, the function $\sigma\overline{{\mathrm{mex}}}(\pi)$ is the sum of the minimal excludants over all overpartitions of $n$

. In this paper, we prove that the $\sigma\overline{{\mathrm{mex}}}(\pi)$ equals the number of partitions of $n$

into distinct parts using three colors. We also provide an asymptotic formula for $\sigma\overline{{\mathrm{mex}}}(\pi)$ and show that $\sigma\overline{{\mathrm{mex}}}(\pi)$ is almost always even and is odd exactly when $n$

is a triangular number. Moreover, we generalize $\overline{{\mathrm{mex}}}(\pi)$ using the least $r$

-gaps, denoted $\overline{{\mathrm{mex}}}_r(\pi)$ , defined as the smallest part of the non-overlined parts of the overpartition $\pi$ appearing less than $r$

times. Similarly, for a positive integer $n$

, the function $\sigma_r\overline{{\mathrm{mex}}}(\pi)$ is the sum of the least $r$

-gaps over all overpartitions of $n$

. We derive a generating function and an asymptotic formula for $\sigma_r\overline{{\mathrm{mex}}}(\pi)$ . Lastly, we study the arithmetic density of $\sigma_r\overline{{\mathrm{mex}}}(\pi)$ modulo $2^k$

, where $r=2^m\cdot3^n, m,n \in \mathbb{Z}_{\geq 0}$ .

A Minimal Excludant over Overpartitions

Victor Manuel R. Aricheta and Judy Ann L. Donato Institute of Mathematics University of the Philippines Diliman Diliman, Quezon City 1101 Philippines

Victor Manuel R. Aricheta and Judy Ann L. Donato
Institute of Mathematics
University of the Philippines Diliman
Diliman, Quezon City 1101
Philippines