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2 pages, integer arithmetic computer proof
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i-matching, r-regular, bipartite
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\title{Extension of Range of Validity of Pernici's 1/n Conjecture}
\author{Paul Federbush \\ Department of Mathematics \\ University of Michigan \\
Ann Arbor, MI 48109-1043 \\
pfed@umich.edu}
\date{\today} 


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\begin{document}

\maketitle
\begin{abstract}

	We understand Pernici's 1/n conjecture to be eqs. (16) and (17) of his paper
"Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy'',
[1]. We have proven a generalized form of eq.(16) in [2]. In this note we extend the
range of validity of (17) from $j\leq100$ , $r\leq10$, shown by computer computation in
[1], with the range $j\leq29$, all $r$, also shown by computer computation. Naturally
we are using the notation of [1].
\end{abstract}
We first recall eqs. (16) and (17) from [1].
\begin{align*}
   [j^k n^{-h}]\,\ln \biggl( 1 + H_j \biggr) =
    [j^k n^{-h}]\,\ln \biggl( 1 + \sum_{s=1}^{j-1} \frac{a_s(r,j)}{n^s} \biggr) &=
  0, \qquad k \ge h+2 \\
  [j^{h+1} n^{-h}]\,\ln \biggl( 1 + H_j \biggr)=  [j^{h+1} n^{-h}]\,\ln \biggl( 1 + \sum_{s=1}^{j-1} \frac{a_s(r,j)}{n^s} \biggr)
  &= \frac{1}{(h+1)h} \biggl( \frac{1}{r^h} - 2 \biggr) 
\end{align*}
We are working with j-matchings on an r-regular bipartite lattice with 2n vertices.
To understand the remaining notation in these equations one need only read
eq.(3) and the text near this equation, and eqs.(10) - (15). all in [1] of course.
After eq.(20) Pernici states he has computer checked these equations for
$j\leq100$ , $r\leq10$. We have proven a generalized form of eq.(16) in [2]. 
For $h\leq23$ we directly computer checked the two sides of the second equation are
equal functions of r. The two sides are polynomials in r of degree h. By substituting
r=1,2,......,h+1 in the two sides and computer checking the two sides are numerically
equal, one verified the equation for $h\leq29$. One is working with rational functions
of all the variables, integer arithmetic on the computer. I worked with Maple on my
desktop computer with 8 M memory. My limiting values were determined by
running out of memory, clearly better equipment and technical improvements
in the program should be expected to extend the limits. The time and memory
use grew surprisingly slowly with h, ( at least I was surprised ). Where are the
good graduate students who should be doing these calculations? Pernici's
paper is fine mathematical physics.













\begin{thebibliography}{1}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


	\bibitem{1} M. Pernici, ``$\frac{1}{n}$ Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy'', J. Stat. Phys \underline{168}, 666 (2017)

	\bibitem{2} P. Federbush, ``A PROOF of Weak Graph Positivity for a Large Range of the Parameters'', arXiv: 2012.10927.

\end{thebibliography}


\end{document}
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