{
  "abstract": "By eliminating the win condition in the game of Connect Four and extending the board to infinite height, a rich state space of positions is obtained. We investigate the number of positions reachable on an $n$-column board after $k$ color-alternating moves. For fixed $k$ we demonstrate polynomiality, derive a partial formula for the polynomial coefficients, and precisely characterize the asymptotic behavior as $n \\to \\infty$. We then turn our attention to the fixed-$n$ case and show that, under a natural addition operation, positions reachable in an even number of moves form a monoid with a highly symmetric finite generating set; by examining certain free submonoids, we bound the exponential growth rate as $k \\to \\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: J. Comb. Math. Comb. Comput. 131 (2026) 57-65",
  "arxiv": "2609.33068",
  "author_total": 1,
  "authors": [
    {
      "name": "Alistair Folster"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/etx2p0pxjrqv8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33068",
  "fulltext_url": null,
  "id": "etx2p0pxjrqv8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33068",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.33068",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/etx2p0pxjrqv8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33068",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33068"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Counting Reachable Positions in a Simplified Model of Connect Four",
  "title_text": "Counting Reachable Positions in a Simplified Model of Connect Four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/etx2p0pxjrqv8",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.61091/jcmcc131-04",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}