The minimal excludant and the least $r$-gaps of $\ell$-regular partitions
The minimal excludant (mex) of an integer partition $λ$ is the smallest positive integer that is not a part of $λ$. In this paper we study an analogue of this statistic for $\ell$-regular partitions with $\ell\geq2$, that is, partitions in which no part is divisible by $\ell$. Recently, Meher \cite{Meher} defined the minimal excludant for $\ell$-regular partitions. Motivated by Meher \cite{Meher}, for an $\ell$-regular partition we define its least $r$-gap as the smallest positive integer not divisible by $\ell$ that appears fewer than $r$ times as a part, and we obtain the two-variable generating function for the number of $\ell$-regular partitions of $n$ with a given least $r$-gap. Let $σ^{\mex}_{\ell}(n)$, and $σ^{\mex}_{\ell,r}(n)$ be the sum of the of the mex values over all $\ell$-regular partitions of $n$ and least $r$-gaps over all $\ell$-regular partitions of $n$ respectively. Using Jacobi's triple product identity, we derive explicit expressions for the generating function of $σ^{\mex}_{\ell}(n)$, and $σ^{\mex}_{\ell,r}(n)$, with separate formulas for even and odd $\ell$. We also show that the sequence $\{σ^{\mex}_{\ell,r}(n)\}_{n\ge0}$ is weakly increasing for every $\ell\ge2$ and $r\ge1$. Also, using Ingham's Tauberian theorem, we establish Hardy--Ramanujan type asymptotic formulas for $σ^{\mex}_{\ell,r}(n)$ when $\ell=4,5,7$ and $r\ge1$ is arbitrary. Lastly, we prove $σ^{\mex}_{\ell,r}$ is odd infinitely often, and we establish a quantitative lower bound, showing that the number of such n up to X grows at least like log log X.