{
  "abstract": "We study classes of permutations that are defined by avoidance of forbidden patterns. For each fixed finite set $B$ of forbidden patterns, we study the \\emph{avoider enumeration problem}: given an integer $n$, output all $B$-avoiding permutations of length $n$. Avoider enumeration is output-polynomial for every fixed $B$. However, our main notion of efficiency is polynomial delay. We use the insert-maximum tree as a search tree, in which permutations are built by repeatedly inserting a new maximum while preserving pattern avoidance. This leads to the \\emph{insert-maximum extension problem}, which asks if a given permutation has an avoiding descendant at a specified depth. An algorithm for this problem can prune branches that cannot reach the target length. Hence, if this problem is in P, we obtain polynomial-delay enumeration. To investigate this problem, we introduce and study \\emph{pockets}, the maximal finite subtrees of the insert-maximum tree. For fixed $B$, if there is a polynomial $p$ such that every pocket rooted at $ρ$ has at most $p(|ρ|)$ nodes, then the extension problem is solvable in polynomial time and enumeration has polynomial delay. In particular, a bound on pocket depth depending only on $B$ suffices. We prove such a bound if the class is well-quasi-ordered, has a regular insertion encoding, or if $B$ omits one of the three maximum-position types: leftmost, interior, or rightmost. We also prove that extension is in NP for every fixed $B$, even with binary-encoded target depth. On the negative side, we construct a fixed set of $55$ forbidden patterns for which extension is NP-complete. Consequently, enumerating valid extensions from a given starting permutation cannot have polynomial delay unless P=NP. This also answers a question of Hanika by showing that the forbidden permutation constraint satisfaction problem can be NP-complete for a fixed forbidden set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37940",
  "author_total": 2,
  "authors": [
    {
      "name": "Manuel Bodirsky"
    },
    {
      "name": "Mohamed Elsheikh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e97hvadb32ytc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37940",
  "fulltext_url": null,
  "id": "e97hvadb32ytc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37940",
  "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",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e97hvadb32ytc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37940"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Enumeration Complexity of Permutation Classes",
  "title_text": "Enumeration Complexity of Permutation Classes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/e97hvadb32ytc",
  "venue": "arXiv",
  "versions": []
}