Determinant identities for the Catalan, Motzkin and Schroder numbers

Abstract

In this paper, we find formulas for the determinants of several Hessenberg matrices whose nonzero entries are derived from the Catalan, Motzkin and Schroder number sequences. By a generalization of Trudi’s formula, we obtain equivalent multi-sum identities involving sums of products of terms from these sequences. We supply both algebraic and combinatorial proofs of our results. For the latter, we draw upon the combinatorial interpretations of the Catalan, Motzkin and Schroder numbers as enumerators of certain classes of first-quadrant lattice paths. As a consequence of our results and the arguments used to establish them, one obtains both new formulas and combinatorial interpretations for some well-known integer sequences, including the central binomial coefficients, grand Motzkin numbers, Delannoy numbers and several entries from the On-Line Encyclopedia of Integer Sequences.