We study multiplicities \(a^{d\lambda }_{\mu ,(dk)}\) of highest weight representations \({\mathbb {S}}_{d\lambda }({\mathbb {C}}^n)\) , \(\lambda \vdash pk\) , of length at most p, in \(\mathbb {S}_{\mu }(S^{dk}({\mathbb {C}}^n))\) , \(\mu \vdash p\) , so called plethysm coefficients, as d tends to \(\infty \) . These are given by quasi-polynomials, which in the case of \(S^p(S^{dk}({\mathbb {C}}^n))\) can explicitly be computed by Pieri’s rule. We show that for all but a finite, explicit list of \(\lambda \) ’s, the leading term is in fact constant and that \(\begin{aligned} a^{d\lambda }_{\mu ,(dk)}\sim \frac{\dim V_\mu }{p!}c^{d\lambda }_{p,dk} \end{aligned}\) as \(d\rightarrow \infty \) . In particular, we answer a conjecture of Kahle and Michałek, going back to Howe.
