{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24848",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrea Ottolini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8z8j5ccs828vm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24848",
  "fulltext_url": null,
  "id": "8z8j5ccs828vm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24848",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7214042669",
  "pdf_url": "https://arxiv.org/pdf/2609.24848",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8z8j5ccs828vm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24848",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24848"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Optimal support and condensation in random allocations",
  "title_text": "Optimal support and condensation in random allocations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8z8j5ccs828vm",
  "venue": "arXiv",
  "versions": []
}