  We derive the limit shape of Young diagrams, associated with growing integer
partitions, with respect to multiplicative probability measures underpinned by
the generating functions of the form $\mathcal{F}(z)=\prod_{\ell=1}^\infty
\mathcal{F}_0(z^\ell)$ (which entails equal weighting among possible parts
$\ell\in\mathbb{N}$). Under mild technical assumptions on the function
$H_0(u)=\ln(\mathcal{F}_0(u))$, we show that the limit shape $\omega^*(x)$
exists and is given by the equation $y=\gamma^{-1}H_0(\mathrm{e}^{-\gamma x})$,
where $\gamma^2=\int_0^1 u^{-1}H_0(u)\,\mathrm{d}u$. The wide class of
partition measures covered by this result includes (but is not limited to)
representatives of the three meta-types of decomposable combinatorial
structures --- assemblies, multisets and selections. Our method is based on the
usual randomization and conditioning; to this end, a suitable local limit
theorem is proved. The proofs are greatly facilitated by working with the
cumulants of sums of the part counts rather than with their moments.
