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

Michael A. Allen, Kenneth Edwards

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.

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

Abstract

By considering the tiling of an -board (a linear array of 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 rectangular sub-tiles of dimensions separated by gaps of width . The interpretation is used to give combinatorial proof of new convolution-type identities relating for the cases , , , for to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are , , and above the leading diagonal. When these identities reduce to ones connecting the Padovan and Narayana's cows numbers.

Paper Structure

This paper contains 6 sections, 11 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Combs
  3. Tiling with half-squares and $(\frac{1}{2},\frac{1}{2};m+2)$-combs where $m\ge0$
  4. Tiling with $(\frac{1}{2},\frac{1}{2};m+1)$- and $(\frac{1}{2},\frac{1}{2};m+2)$-combs where $m\ge1$
  5. Identities relating the Narayana's cows and Padovan numbers
  6. Discussion

Key Result

Theorem 1.1

If $W=\{-1,d_1,\ldots,d_r\}$ where $0\le d_1<d_2<\cdots<d_r$ and $d_r>0$ then for all $n\ge0$, where we take $P^W_{n<0}=0$ and $\delta_{i,j}$ is 1 if $i=j$ and 0 otherwise.

Figures (5)

  • Figure 1: Examples of mixed metatiles and their symbolic representations when tiling with (a) half-squares ($h$) and $(\frac{1}{2},\frac{1}{2};m+2)$-combs ($C$) (b) $(\frac{1}{2},\frac{1}{2};m+1)$-combs ($c$) and $C$ when $m=1$. Each upper (lower) vertical line represents a tooth filling a left (right) slot. The horizontal lines show which teeth are part of the same comb. Interior tiles are indicated by thicker lines. The first two metatiles in each case are examples of pairs of metatiles in the sense of Lemma \ref{['L:mtpairs']}.
  • Figure 2: Digraphs for generating configurations of interior tiles when tiling with half-squares ($h$) and $(\frac{1}{2},\frac{1}{2};m+2)$-combs ($C$).
  • Figure 3: Digraphs for tiling using $F_m$ ($F$), $\bar{F}_1$ ($\bar{f}$), and $\bar{F}_0$ ($\bar{S}$). The $\bar{0}^{m/2}1$ node is only present if $m$ is even.
  • Figure 4: Digraphs for generating configurations of interior combs when tiling with $(\frac{1}{2},\frac{1}{2};m+1)$-combs ($c$) and $(\frac{1}{2},\frac{1}{2};m+2)$-combs ($C$).
  • Figure 5: Digraphs for tiling using $F_m$ ($F$), $F_{m-1}$ ($f$), and $\bar{F}_1$ ($\bar{f}$).

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 4.1
  • Corollary 4.2
  • Lemma 4.3
  • ...and 1 more