Combinatorial Identities for Incomplete Tribonacci Polynomials

Publication date: 10 June 2014

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).

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Combinatorial identities, bijective combinatorics (05A19) Fibonacci and Lucas numbers and polynomials and generalizations (11B39)

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