Hooks and Hockey Sticks on the Catalan Triangle

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.