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Peer-reviewedCC BY 4.0Mathematics

Hooks and Hockey Sticks on the Catalan Triangle

  • Ahmet Zahid Küçük1
  1. 1Department of Engineering Basic Sciences, Faculty of Engineering and Natural Sciences, Konya Technical University, 42130 Konya, Türkiye
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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.

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Cite

BibTeXRIS
BibTeX
@article{kucuk2026hooks,
  title = {{Hooks and Hockey Sticks on the Catalan Triangle}},
  author = {Ahmet Zahid Küçük},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14193609},
  url = {https://www.mdpi.com/2227-7390/14/19/3609},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Hooks and Hockey Sticks on the Catalan Triangle
AU  - Ahmet Zahid Küçük
PY  - 2026
DA  - 2026/10/05
JO  - Mathematics
DO  - 10.3390/math14193609
UR  - https://www.mdpi.com/2227-7390/14/19/3609
AB  - 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.
SN  - 2227-7390
ER  -