Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| author | Allen, Michael A. Edwards, Kenneth |
| author_facet | Allen, Michael A. Edwards, Kenneth |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_02589 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices Allen, Michael A. Edwards, Kenneth Combinatorics Primary 05A19, Secondary 05A05, 11B37, 11B39 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. |
| title | Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices |
| topic | Combinatorics Primary 05A19, Secondary 05A05, 11B37, 11B39 |
| url | https://arxiv.org/abs/2107.02589 |