Determinants of some Hessenberg–Toeplitz matrices with Motzkin number entries

Abstract

In this paper, we find formulas for the determinants of some Hessenberg–Toeplitz matrices whose nonzero entries are derived from the Motzkin number sequence and its translates. We provide both algebraic and combinatorial proofs of our results, making use of generating functions for the former and various counting methods, such as direct enumeration, sign-changing involutions, and bijections, for the latter. In the process, it is shown that three important classes of lattice paths—namely, the Motzkin paths, the Riordan paths, and the so-called Motzkin left factors—have their cardinalities given as determinants of certain Hessenberg–Toeplitz matrices with Motzkin number entries. Further formulas are found for determinant identities involving two sequences from the On-Line Encyclopedia of Integer Sequences, which are subsequently explained bijectively.