Combinatorial proofs of identities for the generalized Leonardo numbers

Abstract

In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by \mathcal{L}_{k,n}, as well as derive some new formulas. To do so, we interpret \mathcal{L}_{k,n} as the enumerator of two classes of linear colored tilings of length n. A comparable treatment is also given for the incomplete generalized Leonardo numbers. Finally, a (p,q)-generalization of \mathcal{L}_{k,n} is obtained by considering the joint distribution of a pair of statistics on one of the aforementioned classes of colored tilings.