Combinatorial Identities for Incomplete Tribonacci Polynomials

TL;DR

This paper offers a combinatorial interpretation of incomplete tribonacci polynomials, enabling new proofs and identities, and derives their generating function using combinatorial methods.

Contribution

It introduces a combinatorial model for incomplete tribonacci polynomials and derives their generating function, expanding understanding and providing tools for further research.

Findings

Combinatorial interpretation via weighted linear tilings.

New identities for incomplete tribonacci polynomials.

Explicit formula for the generating function.

Abstract

The incomplete tribonacci polynomials, denoted by T_n^{(s)}(x), generalize the usual tribonacci polynomials T_n(x) and were introduced in [10], where several algebraic identities were shown. In this paper, we provide a combinatorial interpretation for T_n^{(s)}(x) in terms of weighted linear tilings involving three types of tiles. This allows one not only to supply combinatorial proofs of the identities for T_n^{(s)}(x) appearing in [10] but also to derive additional identities. In the final section, we provide a formula for the ordinary generating function of the sequence T_n^{(s)}(x) for a fixed s, which was requested in [10]. Our derivation is combinatorial in nature and makes use of an identity relating T_n^{(s)}(x) to T_n(x).