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

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 (where , , , where and are positive integers and ) each raised to an arbitrary non-negative integer power. A -comb is a tile composed of m rectangular sub-tiles of dimensions separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating for the cases q = 2, , , for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are , M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers.