Optimal support and condensation in random allocations

Abstract

How many distinct symbols should a password use? If its length is fixed at $n$ and an observer learns only which symbols appear, the number of compatible passwords is maximized asymptotically when $k/n\to1/(2\log2)$. We ask what changes when, in addition to the length, aggregate information about the repetition pattern is revealed. We model this by fixing a second additive profile $V_k=\sum_i v(J_i)$ at scale $V_k/n\approxρ$. For $v(j)=\log(j!)$, the profile records the reduction in the logarithm of the number of compatible words caused by repetitions; we show that once the normalized profile $ρ$ exceeds $0.507834\ldots$, the limiting optimal fraction is pinned at $1/2$. For a typical multiplicity profile at the optimal support above this threshold, the excess in $V_k$ is carried by a vanishing fraction of used symbols. We interpret this as a form of non-equivalence of ensembles and extend the mechanism to other profiles and non-uniform allocation models.