Combinatorial Identities for Incomplete Tribonacci Polynomials
Combinatorics
2014-06-12 v1
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).
Keywords
Cite
@article{arxiv.1406.2755,
title = {Combinatorial Identities for Incomplete Tribonacci Polynomials},
author = {Mark Shattuck},
journal= {arXiv preprint arXiv:1406.2755},
year = {2014}
}