Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices

TL;DR

This paper provides a combinatorial interpretation of the product of generalized Fibonacci numbers squared using tiling with comb-shaped tiles, and relates these to permanents of specific Toeplitz matrices, connecting Fibonacci identities to matrix theory.

Contribution

It introduces a novel combinatorial tiling interpretation for squared generalized Fibonacci numbers and establishes new identities linking these to Toeplitz matrix permanents.

Findings

Combinatorial proof of identities involving squared generalized Fibonacci numbers.

Connection between Fibonacci number products and permanents of (0,1) Toeplitz matrices.

Reduction of identities to known Fibonacci-related sequences like Padovan and Narayana's cows numbers.

Abstract

By considering the tiling of an $N$-board (a linear array of $N$ square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers $s_n$ (where $s_{n}=\sum_{i=1}^q v_i s_{n-m_i}$, $s_0=1$, $s_{n<0}=0$, where $v_i$ and $m_i$ are positive integers and $m_1<\cdots<m_q$) each raised to an arbitrary non-negative integer power. A $(w,g;m)$-comb is a tile composed of $m$ rectangular sub-tiles of dimensions $w\times1$ separated by gaps of width $g$. The interpretation is used to give combinatorial proof of new convolution-type identities relating $s_n^2$ for the cases $q=2$, $v_i=1$, $m_1=M$, $m_2=m+1$ for $M=0,m$ to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are $-2$, $M-1$, and $m$ above the leading diagonal. When $m=1$ these identities reduce to ones connecting the Padovan and Narayana's cows numbers.