{
  "abstract": "The cardinalities of Kiselman's semigroups $K_n$ are known to grow double exponentially in $n$: the sequence $2^{-n/2}\\log_2|K_n|$ converges along the even and odd $n$ subsequences to positive limits $c_{\\mathrm{even}}$ and $c_{\\mathrm{odd}}$ that appear to coincide. In this paper we prove that they are indeed distinct. Since Kiselman's semigroups are $\\mathcal{L}$-trivial, the suffix relation defines a partial order on $K_n$. We introduce the \\emph{Kiselman transform} $\\mathsf{Kis}(P)$ of a poset $P$, a construction recovering the poset associated with $K_{n+1}$ from that associated with $K_n$. Counting strictly increasing chains in $P$ and $\\mathsf{Kis}(P)$ yields a linear recurrence governed by a symmetric infinite matrix with explicit binomial entries. This can be used towards efficiently computing $|K_n|$, allowing the values listed in OEIS sequence A125625 to be extended beyond $n=14$. A generating functions analysis gives formulas for the above asymptotic limits, together with rigorous numerical bounds establishing that \\[ c_{\\mathrm{even}} - c_{\\mathrm{odd}} = 1.9945066605\\ldots\\times10^{-6} \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14331",
  "author_total": 2,
  "authors": [
    {
      "name": "Alessandro D'Andrea"
    },
    {
      "name": "Salvatore Stella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h1av2ywayrg2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14331",
  "fulltext_url": null,
  "id": "6h1av2ywayrg2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14331",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221783131",
  "pdf_url": "https://arxiv.org/pdf/2610.14331",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h1av2ywayrg2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14331",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14331"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cardinalities of Kiselman monoids and their parity asymptotics",
  "title_text": "Cardinalities of Kiselman monoids and their parity asymptotics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6h1av2ywayrg2",
  "venue": "arXiv",
  "versions": []
}