Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices
1
Issued Date
2022-01-01
Resource Type
ISSN
03081087
eISSN
15635139
Scopus ID
2-s2.0-85135572997
Journal Title
Linear and Multilinear Algebra
Rights Holder(s)
SCOPUS
Bibliographic Citation
Linear and Multilinear Algebra (2022)
Suggested Citation
Allen M.A., Edwards K. Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices. Linear and Multilinear Algebra (2022). doi:10.1080/03081087.2022.2107979 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/85120
Title
Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices
Author(s)
Author's Affiliation
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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 (Formula presented.) (where (Formula presented.), (Formula presented.), (Formula presented.), where (Formula presented.) and (Formula presented.) are positive integers and (Formula presented.)) each raised to an arbitrary non-negative integer power. A (Formula presented.) -comb is a tile composed of m rectangular sub-tiles of dimensions (Formula presented.) separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating (Formula presented.) for the cases q = 2, (Formula presented.), (Formula presented.), (Formula presented.) for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are (Formula presented.), M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers.