{
  "abstract": "We count solutions of composition identities for maps on finite sets, first for three identities in which both maps vary and then for commutation with a prescribed arithmetic map. For $f\\circ g=f\\circ g\\circ g$, drawing an arrow from each element to its image reduces the enumeration to rooted trees attached to directed cycles. For $f\\circ f=f\\circ g\\circ f$, grouping elements with the same $f$-value leads instead to a partition-based enumeration. A classical rooted-forest enumeration leads in both problems to $\\mathcal T(x)$, the exponential generating function for rooted labeled trees. For $f=g\\circ f\\circ g$, the values on the cycles of $g$ determine all remaining values. The exponential generating function $\\mathcal D$ for these ordered pairs and the ordinary generating function $\\mathcal A$ for endofunctions up to relabeling satisfy $\\mathcal D(x)=\\mathcal A(\\mathcal T(x))$. Finally, we count functions on $\\mathbb Z/n\\mathbb Z$ satisfying $f(ax)=af(x)$, first for prime multipliers $a$ and then for a broader class of composite multipliers. In each problem, fixing one map exposes independent choices for the other through its functional digraph or its fibers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08674",
  "author_total": 1,
  "authors": [
    {
      "name": "Max A. Alekseyev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bk3q2m5qdbrev.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08674",
  "fulltext_url": null,
  "id": "bk3q2m5qdbrev",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08674",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08674",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bk3q2m5qdbrev.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08674",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08674"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting Finite Maps through Functional Digraphs and Fibers",
  "title_text": "Counting Finite Maps through Functional Digraphs and Fibers",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/bk3q2m5qdbrev",
  "venue": "arXiv",
  "versions": []
}