{
  "abstract": "Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\\bigcup_{n\\ge 0}D_n$, and let $area, dinv, bounce \\colon D\\to\\mathbb{N}$ be the standard statistics. Then, \\[ C_n(q,t)=\\sum_{P\\in D_n}q^{area(P)}t^{bounce(P)} =\\sum_{P\\in D_n}q^{dinv(P)}t^{area(P)}. \\] Haglund's zeta map $ζ\\colon D\\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $ζ$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a \\emph{rank sort} and introduce \\emph{weighted-rank polyregular maps} (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $ζ^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $ζ$ cannot be extended within WRP to exchange the two statistics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05005",
  "author_total": 4,
  "authors": [
    {
      "name": "Jineon Baek"
    },
    {
      "name": "Byung-Hak Hwang"
    },
    {
      "name": "Joonhyun La"
    },
    {
      "name": "Hongseok Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cc0jgkqgzqnay.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05005",
  "fulltext_url": null,
  "id": "cc0jgkqgzqnay",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05005",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.FL"
    ],
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7211919561",
  "pdf_url": "https://arxiv.org/pdf/2609.05005",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cc0jgkqgzqnay.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05005",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05005"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry",
  "title_text": "A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the q,t-Catalan Symmetry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/cc0jgkqgzqnay",
  "venue": "arXiv",
  "versions": []
}