{
  "abstract": "The inverse of the coefficient matrix of the Chebyshev S-polynomials is the Catalan triangle. From this inverse matrix, we derive rules in the spirit of Pascal’s rule in three distinct senses. First, there is a local hook rule: each entry is generated from two entries one row up, and a similar local rule governs the symmetric matrix SST. Second, there is a cumulative rule: an exact analogue of the classical hockey stick identity, in which a sum of consecutive entries along an anti-diagonal telescopes to a single entry of the matrix. Third, there is a row-by-row relation: the row polynomials of the inverse matrix satisfy a linear recurrence weighted by Catalan numbers, informally a symbolic inverse of the original recurrence. The same hook rule also yields a Catalan-weighted identity linking the rows of the inverse matrix to the S-polynomials.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Engineering Basic Sciences, Faculty of Engineering and Natural Sciences, Konya Technical University, 42130 Konya, Türkiye"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Ahmet Zahid Küçük",
      "orcid": "0000-0001-6816-6478"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c3gm8fcsbtyqd.bib",
  "cited_by": 0,
  "doi": "10.3390/math14193609",
  "fulltext_url": null,
  "id": "c3gm8fcsbtyqd",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/19/3609",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220428546",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/c3gm8fcsbtyqd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14193609",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/19/3609"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Advanced Combinatorial Mathematics",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hooks and Hockey Sticks on the Catalan Triangle",
  "title_text": "Hooks and Hockey Sticks on the Catalan Triangle",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/c3gm8fcsbtyqd",
  "venue": "Mathematics",
  "versions": []
}