Abstract
In this article, we introduce a family of weighted lattice paths, whose step set is $\{H=(1,0), V=(0,1), D_1=(1,1), \dots, D_{m-1}=(1,m-1)\}$. Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the $k$-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and tribonacci sequence, respectively. From this family of Riordan arrays we introduce a generalized $k$-bonacci polynomial sequence, and we give a lattice path combinatorial interpretation of these polynomials. In particular, we find a combinatorial interpretation of tribonacci and tribonacci-Lucas polynomials.
Highlights
A lattice path Γ in the xy-plane with steps in a given set S ⊂ Z2 is a concatenation of directed steps of S, i.e., Γ = s1s2 · · · sl, where si ∈ S for 1 i l
We introduce a family of weighted lattice paths, whose step set is {H = (1, 0), V = (0, 1), D1 = (1, 1), . . . , Dm−1 = (1, m − 1)}
We define a family of Riordan arrays whose sum on the rising diagonal is the k-bonacci sequence
Summary
This array is called Delannoy or tribonacci matrix It satisfies that the sum of the elements on the rising diagonal is the tribonacci sequence tn (sequence A000073). The first few terms of this triangle are 0 They introduced a generalized Lucas polynomial sequence, and they gave a lattice path combinatorial interpretation for these polynomials. A natural questions is: What family of steps are required to find a Riordan array, such that the sum of the elements on its rising diagonal is the k-bonacci numbers Fn(k)? From the number of these weighted lattice paths, we obtain a new family of Riordan arrays, such that the sum of the elements on its rising diagonal is the k-bonacci sequence. The new family of weighted lattice paths leads us to a combinatorial interpretation for the generalized k-bonacci polynomial sequence. We show three combinatorial interpretations of the central tetrabonacci numbers
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