%rna.tex: a Plain TeX file by AJ Bu, Manuel Kauers and  and Doron Zeilberger
%Statistical Analysis of Hairpins and BasePairs in RNA Secondary Structures

%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
  0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
  height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\S{{\cal S}}
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\W{{\cal W}}
\def\1{{\overline{1}}}
\def\2{{\overline{2}}}
\parindent=0pt
\overfullrule=0in
\def\Tilde{\char126\relax}
\def\frac#1#2{{#1 \over #2}}
%\headline={\rm  \ifodd\pageno  \RightHead  \else  \LeftHead  \fi}
%\def\RightHead{\centerline{
%Title
%}}
%\def\LeftHead{ \centerline{Doron Zeilberger}}
%end macros
\centerline
{\bf
Statistical Analysis of Hairpins and BasePairs in RNA Secondary Structures
}
\bigskip
\centerline
{\it AJ BU , Manuel KAURES, and Doron ZEILBERGER}
\bigskip

\qquad {\it In memory of Kequan Ding (?-2025) }

{\bf Abstract}: We find precise asymptotic expressions for the
expectations, variances, covariance, and quite a few further mixed moments for 
number of hairpins and basepairs in RNA secondary structure and give convincing
evidence that the central-scaled distribution of the pair of random variables
(hairpins,basepairs) tends in distribution to the bi-variate normal distribution
with correlation  $\sqrt{5 \sqrt{5} -11}/2= 0.2123322205\dots$


{\bf Theorem}: Let $X_n$ and $Z_n$ be the discrete random variable, defined on the set of RNA secondary structures.
We have the following asmptotic expressions
$$
\eqalign{
E[X_n]&= (1-\frac{2}{5}\sqrt{5})n \cdot \Bigl(1+(\frac{7}{4}+\frac{11}{20}\sqrt{5})n^{-1}+(\frac{243}{160}+\frac{99}{160}\sqrt{5})n^{-2}+(\frac{339}{160}+\frac{1029}{800}\sqrt{5})n^{-3}\cr
      &+(\frac{11917563}{819200}+\frac{5400687}{819200}\sqrt{5})n^{-4}+{\rm O}(n^{-5})\Bigr)\cr
E[Z_n]&= (\frac{1}{2}-\frac{1}{10}\sqrt{5})n \cdot \Bigl(1+(-\frac{5}{8}-\frac{13}{40}\sqrt{5})n^{-1}+(\frac{21}{320}+\frac{21}{320}\sqrt{5})n^{-2}\cr
      &+(-\frac{93}{320}-\frac{177}{1600}\sqrt{5})n^{-3}+(-\frac{13887249}{1638400}-\frac{6272793}{1638400}\sqrt{5})n^{-4}+{\rm O}(n^{-5})\Bigr)\cr
Var[X_n] &= (2-\frac{22}{25}\sqrt{5})n \cdot \Bigl(1+(\frac{1}{16}+\frac{23}{80}\sqrt{5})n^{-1}+(-\frac{651}{160}-\frac{177}{64}\sqrt{5})n^{-2}\cr
     &+(-\frac{1208783}{81920}-\frac{3216693}{409600}\sqrt{5})n^{-3}+{\rm O}(n^{-4})\Bigr)\cr
Var[Z_n] &= (0+\frac{1}{50}\sqrt{5})n \cdot \Bigl(1+(1+\frac{1}{10}\sqrt{5})n^{-1}-\frac{261}{160}n^{-2}\cr
     &+(-\frac{27179}{2560}-\frac{915879}{204800}\sqrt{5})n^{-3}+{\rm O}(n^{-4})\Bigr)\cr
Corr[X_n,Z_n] &= \frac{1}{2} \sqrt{5\sqrt{5}-11}\cdot \Bigl(1+(\frac{15}{32}+\frac{13}{32}\sqrt{5})n^{-1}+(\frac{4469}{1024}+\frac{1345}{1024}\sqrt{5})n^{-2}\cr
     &+(\frac{1766711}{32768}+\frac{801983}{32768}\sqrt{5})n^{-3}+{\rm O}(n^{-4})\Bigr)
}
$$

We have  lots of evidence that the following conjecture is true, and one of us (DZ) is offering a donation of \$100 to the OEIS for its rigorous proof

{\bf Conjecture}: The centralized-scaled version of the pair $(X_n,Z_n)$, namely
$$
 \left (  \frac{ X_n-E[X_n] }{\sqrt{Var(X_n)}}, \frac{Z_n-Z[X_n]}{\sqrt{Var(Z_n)}}  \right ) \quad ,
$$
tends, in distribution, to the bi-variate normal distribution with covariance
$c=\frac{\sqrt{5 \sqrt{5} -11}}{2}$, whose probability density function (pdf) is:
$$
\frac{{ e}^{-\frac{1}{2} x^{2}-\frac{1}{2} y^{2}+c x y} \sqrt{-c^{2}+1}}{2 \pi} \quad .
$$




{\bf References}

[Sl] Neil A. J. Sloane, {\it The On-Line Encyclopedia of Integer Sequences® (OEIS)}, {\tt https://oeis.org/}.



\bigskip
\hrule
\bigskip
AJ Bu,  Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt   ab1854  at math dot rutgers dot edu}
\bigskip
Manuel Kauers, Institute for Algebra, Johannes Kepler Universit\"at, Altenbergerstra\ss e 69 A-4040 Linz, Austria
Email: {\tt   manuel dot kauers at jku dot at}
\bigskip
Doron Zeilberger,  Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt   ab1854  at math dot rutgers dot edu}


Written: {\bf  Jan. 30, 2026}.

\end


