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%\LRH{Moderate Growth Time Series For Dynamic Combinatorics modelisation}

%\RRH{Lua\"\i Jaff, Hatem Haje-Kacem, G\'erard H.E. Duchamp and
%Cyrille Bertelle}

%\VOL{x}

%\ISSUE{x/y/z}

%\PUBYEAR{2006}

\setcounter{page}{1}

%\LRH{Moderate Growth Time Series For Dynamic Combinatorics modelisation}

%\RRH{Lua\"\i Jaff, Hatem Haje-Kacem, G\'erard H.E. Duchamp and
%Cyrille Bertelle}

%\VOL{x}

%\ISSUE{x/y/z}

%\PUBYEAR{2006}

%\BottomCatch


\title{\textbf{MODERATE GROWTH TIME SERIES 
FOR \\DYNAMIC COMBINATORICS MODELISATION}}


\author{
%\texbf{
\textbf{Lua\"\i ~JAFF, G\'erard H.E. DUCHAMP},\\
{\it \textbf{LIPN, Paris XIII University}}\\
{\it \textbf{99, avenue Jean-Baptiste Cl\'ement}} \\
{\it \textbf{93430 Villetaneuse, France}} \\
\normalsize {\tt luai.jaff@gmail.com, gheduchamp@gmail.com} \\~\\
%
\textbf{Hatem HADJ KACEM},\\
{\it \textbf{MIRACL, Sfax University}}\\
{\it \textbf{Route de l'Aérodrome - BP 559}}\\
{\it \textbf{3029 Sfax, Tunisia}}\\
\normalsize{\tt hatem.hadjkacem@fsegs.rnu.tn} \\~\\
%
\textbf{Cyrille BERTELLE},\\
{\it \textbf{LITIS, Le Havre University}}\\
{\it \textbf{25 rue Ph. Lebon - BP 540}}\\
{\it \textbf{76058 Le Havre Cedex, France}}\\
\normalsize{\tt cyrille.bertelle@univ-lehavre.fr}
%}
}

\date{}

\begin{document}


\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{INTRDUCTION}
%In 1962 Schensted was discovered a bijective relation between the permutations on [n],
%on the one hand, and the pair of Young tableaux of the same form, on the other, known under
%the name of the correspondance of Robinson-Schensted.
%The study of the combinatorial famillies like permutations, tableaux
%and Dyck's words need to find the connection between them.
%In the present work, we determinate the bijection theory that allow us to go
%from codes to permutations of symmetric group, Young tableaux of heigth two and Dyck word.
%In the other hand, we can see our codes as an economic model, this is meaning that the codes
%are the behaviour of a person which win money every month without loss his capital.
%Firstly, in the first section we talk about ...

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

%\begin{abstract}
\section*{Abstract}
Here, we present a family of time series with a simple growth constraint.
This family can be the basis of a model to apply to emerging computation
in business and micro-economy where global functions can be expressed from local rules.
We explicit a double statistics on these series which allows to establish a
one-to-one correspondence between three other ballot-like strunctures.\\

%\end{abstract}


{\bf Keywords:} Time series; Complex system; Growth constraint; local rules; Dyck words; Permutations; Codes.


\section{Introduction}

In this paper, we are interested with time series with moderate growth but possibly sudden decay.
%\\{\bf todo ruine du rentier}\\ 
We will focus ourselves on a very simple model (a ``toy model'' as physicists may say),
the combinatorics of which is completely mastered.
This feature is important as one may use simulations and estimates over ``all the possible configurations'',
as it is the case, for example, for other combinatorial models (Cox-Ross-Rubinstein, for instance).
The model is that of sequences with integer values and growth bounded by a unit (local rule).
Surprisingly, there is one-to-one correspondences between the possible configurations and planar
combinatorial objects which are endowed with a special dynamics which we describe here.
The structure of the paper is the following. Section 2 presents an applicative economic problem which leads to generate the studied growth time series from local rules. On section 3, we propose a non exhaustive review concerning emerging computation in economic domain and how our work relates corresponding of this body of knowledge. Section 4 develops the dynamic combinatorics computation which leads to establish one-to-one correspondences between three other ballot-like structures. We conclude on section 5.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{From Micro-Economy Local Rules To Dynamic Combinatorics}

Our aim is to describe here a {\bf toy-model} of the benefit in the
following situation.
% \newline
A capital owner possesses two accounts,
say {\bf P} and {\bf R}, {\bf P} is the account where the principal
(untouched) capital is deposited. This capital produces a constant
return (one unit per unit of time) which is sent to a reserve {\bf R}.
From the account {\bf R} can be with drawn arbitrary amounts of money and the
account must stay positive.

%\begin{figure}[htbp]
%\begin{center}
%\input{a1.pstex_t}
%\caption{accounts P and R }
%\end{center}
%\end{figure}

\noindent The possible configurations are described the sequences
such that


\begin{itemize}
\item $a_1 = 0$
\item $a_{i+1} \leq a_i + 1$
\end{itemize}

%\vspace{1cm}
%\begin{center}
\begin{figure} [htp]
\begin{center}
\setlength{\unitlength}{1.2mm} 
\begin{picture}(75,50)
\thinlines\put(0,0){\vector(1,0){45}}
\put(0,0){\vector(0,1){45}}
\put(1,46){$\mathbf{R}$}
\put(45,-3){$t$}
\linethickness{0.075mm}
\multiput(5,0)(5,0){8}{\line(0,1){40}}
\multiput(0,5)(0,5){8}{\line(1,0){40}}

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%\thicklines
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%\dottedline[$\star$]{5}(0,0)(5,5)(10,10)(15,15)(20,20)(25,25)(30,30)(35,35)(40,40)
\dottedline[$\bullet$]{10}(0,0)(10,10)(20,20)(30,30)(40,40)
\put(42,40){$012345678$}
\put(0,0){\line(1,0){40}}
\dottedline[$\bullet$]{5}(0,0)(40,0)
\put(42,1){$000000000$}
\drawline[1](0,0)(5,5)(15,5)(25,15)(30,0)(40,10)
\put(42,10){$011123012$}
\drawline[1](0,0)(10,10)(15,0)(35,20)(40,15)
\put(42,15){$012012343$}
%\put(20,15){\line(1,0){10}}
%\put(30,15){\line(0,1){5}}

\put(-3,0){\texttt{0}}
\put(-3,5){\texttt{1}}
\put(-3,10){\texttt{2}}
\put(-3,15){\texttt{3}}
\put(-3,20){\texttt{4}}
\put(-3,25){\texttt{5}}
\put(-3,30){\texttt{6}}
\put(-3,35){\texttt{7}}
\put(-3,40){\texttt{8}}
%\put(10,35){
%\begin{minipage}[b]{50mm}
%\footnotesize Transmission t\'etravalente.\\ 
%Vitesse de modulation   :   $1/T$ bauds.\\ 
%D\'ebit binaire   :   $2/T$ bits/s.
%\end{minipage}}
%%%%%\put(2,-15){
%\begin{minipage}[b]{70mm}
%\footnotesize 
%\end{minipage}}
\end{picture}
\caption{Maximal, minimal (dotted)
and two intermediate trajectories.
Their codes are on the right.}
\end{center}
\end{figure}
%\end{center}

% % % \begin{figure}[htbp]
% % % \begin{center}
% % % %\input{a2.pstex_t}
% % % %\caption{Maximal M, minimal m and two intermediate trajectories
% % % %(t1,t2)}
% % % \end{center}
% % % \end{figure}

\hspace{2cm}

In this paper, we build combinatorial structures that allow to modelize and to compute the global behavior of the reserve {\bf R} by some specific functions. We can consider this result as an emergent function from the basic local rules. 

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

\section{Emerging Computations}

Emerging computation is nowadays a thrilling topic which concerns many developments in complex systems modeling. A brief review can allow to classify these emerging computations concerning economic domains in 3 spaces. \\

The first space is composed of emerging computations which lead to some universal laws. Per Bak's sand pile is concerned by this class \cite{bak}. In such model called Self-Organized Criticality, the phenomenon is crossed by transformation which make it evolve by avalanche. The Coton market trade follows such a law. For 1000 small price variations, there are only 100 middle price variations and only 10 major price variations. The general law which characterize such criticality phenomena is an exponential law.\\

The second space of emerging computation leads to some pattern formations without a complete knowledge of any law. Thomas Schelling's segregation model for urban development is concerned by this class \cite{schelling}. In such model some local interaction between neighbours can lead to self-organized patterns which emerge from the whole interaction systems. Some areas become specialized to some people categories while other areas are devoted to others ones.\\

The third space of emerging computation described here, leads to some global functions expressions. It is typically what we will describe in our problem. The local rules concerned by the proposed economic toy-model will lead to define combinatorics structures allowing to compute a functional global approach. The detailed computation is describe in the following  


%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Dynamics Combinatorics Computation}

\subsection{Trajectories and Codes}
We can define the trajectories of our model by sequences (codes) $a_1 a_2 a_3 ... a_n$ such that 

\begin{itemize}

\item $a_1 = 1$
\item $a_{j+1} \leq a_j + 1$

\end{itemize}

{\bf Example :} For n = 4, we have 14 codes as described in 
the following table.
%figure \ref{codeTableau}

%\begin{figure}
\begin{center}
$$
\begin{tabular}{|c|c|}
\hline
numbers&codes\\
\hline
1 & 1111\\ \hline 2 & 1112\\ \hline 3 & 1121\\ \hline 4 & 1122\\ \hline 5 & 1123\\
\hline 6 & 1211\\ \hline 7 & 1212\\ \hline 8 & 1221\\
\hline 9 & 1222\\ \hline 10 & 1223\\ \hline 11 & 1231\\
\hline 12 & 1232\\ \hline 13 & 1233\\ \hline 14 & 1234\\
\hline

\end{tabular}
$$
%\caption{Codes tableau }
%\label{codeTableau}
\end{center}
%\end{figure}

\noindent
We remark that we have 5 codes which end by 1 or 2,
3 codes ending by 3 and one code ending by 4.
Now if one sets $l(n,k)$ to be the number of codes ending by $k-1$, one can check that

\begin{itemize}
\item $l(n,0) =  l(0,n)=0\ (\forall n\geq 1)$
\item $l(0,0)=1$ (the void sequence) 
\item $l(n,k) = \sum_{j\geq {k+1}} l(n-1,j)$
\end{itemize}

whence the easy computed table of the first values

%\begin{figure}
\begin{center}
\begin{tabular}{||c||c|c|c|c|c|c|c|c||}
\hline
n$\backslash$ k & 0&1&2&3&4&5&6&7\\
\hline
\hline
0&1&0&0&0&0&0&0&0\\
\hline
1& 0&1&0&0&0&0&0&0\\
\hline
2& 0&1&1&0&0&0&0&0\\
\hline
3& 0&2&2&1&0&0&0&0\\
\hline
4& 0&5&5&3&1&0&0&0\\
\hline
5& 0&14&14&9&4&1&0&0\\
\hline
6& 0&42&42&28&14&5&1&0\\
\hline
7& 0&132&132&90&48&20&6&1\\
\hline
\end{tabular}
\end{center}

\medskip
\noindent
The values for $ n, k \geq 1$ can be even more easily computed with the (subdiagonal) local rule
described by \textcolor{red}{\bf West} + \textcolor{red}{\bf North} = \textcolor{black}{\bf result}.
For instance, we remark that \textcolor{red}{\bf 9} + \textcolor{red}{\bf 5} = \textcolor{black}{\bf }14. 

\bigskip
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}

%n$\backslash$ k & 0&1&2&3&4&5&6\\
%\hline
%\hline
%0& 1&0&0&0&0&0&0&0\\
\hline
\textcolor{blue}{\bf 1}&0&0&0&0&0&0\\
\hline
1&\textcolor{blue}{\bf 1}&0&0&0&0&0\\
\hline
1&2&\textcolor{blue}{\bf 2}&0&0&0&0\\
\hline
1&3&5&\textcolor{blue}{\bf 5}&0&0&0\\
\hline
1&4&\textcolor{red}{\bf 9}&14&\textcolor{blue}{\bf 14}&0&0\\
\hline
1&\textcolor{red}{\bf 5}&\textcolor{black}{\bf 14}&28&42&\textcolor{blue}{\bf 42}&0\\
\hline

\end{tabular}
\end{center}

\medskip
\noindent
Remark that the preceding table gives the mirror images of the lines of the previous
double statistics.

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Permutations}

We say that a permutation $\pi$ of n letters has an increasing subsequences of length $k$ if there
are positions $$ 1 \leq i_1 < i_2 < i_3 < ... < i_k \leq n$$ such that
$$\pi(i_1) < \pi(i_2) < \pi(i_3) < ... < \pi(i_k)$$

For example

$$\pi=\left(\begin{array}{ccccc}
1&2&3&4&5 \\ 5&3&4&1&2 \end{array} \right)$$

has increasing subsequences of length 2, at points $\{2,3\}$ as well as at positions
$\{4,5\}$. Let $\pi_2(n)$ be the number of permutations of n letters that have no
increasing subsequences of length $> 2$. By direct enumeration we obtain the following
table.


$$
\begin{tabular}{|c|c|}
\hline
n &$\pi_2(n)$\\
\hline 0 & 1\\ \hline 1 & 1\\ \hline 2 & 2\\ \hline 3 & 5\\
\hline 4 & 14\\ \hline 5 & 42\\ \hline 6 & 132\\ \hline 7 & 429\\
\hline 8 & 1430\\
\hline
\end{tabular}
$$

\begin{proposition}
$|\pi_2(n)| = C_n$, where $\pi_2(n)$  the number of permutations of n letters that have no
increasing subsequences of length $> 2$ and $C_n$ is the n-th Catalan number.
$$C_n = \frac{1}{n+1}  \left(\begin{array}{c}
2n\\ n \end{array} \right)$$

\end{proposition}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Young Tableaux}

\begin{definition}

A {\bf partition} of n, written $\lambda \vdash n$, is a sequence,
$$\lambda = (\lambda_1, \lambda_2, \lambda_3, ... , \lambda_k)$$
such that the $\lambda_i$ are decreasing (weakly) and $\sum_{i=1}^{k} \lambda_i = n$.

Let $\lambda = (\lambda_1, \lambda_2, \lambda_3, ... , \lambda_k) \vdash n$. Then the {\bf Ferrers diagram},
or {\bf shape}, of $\lambda$ is an array of n-squares into k left-justified rows with row $i$ containing
$\lambda_i$ squares for $1 \leq i \leq k$.
For example, the partition (4,3,1)


\begin{figure}[htbp]
\begin{center}\input{r2.pstex_t}
\end{center} \end{figure}


Let $\lambda$ be as above. A {\bf Young tableau} of shape $\lambda$, is an array
obtained by replacing the squares of the shape of $\lambda$ by a bijection with
the numbers 1,2,...,n.

A tableau $T$ is said to be a {\bf standard Young tableau} if the rows and columns are
increasing sequences. For example below the tableau is standard


\begin{figure}[htbp]
\begin{center}\input{TS_4.pstex_t}
\end{center} \end{figure}

A {\bf standard Young tableau of two lines} is of the shape $\lambda(l_1,l_2) \vdash n$ where
$l_1 \geq l_2 > 0$. Let {\bf $f^{(l_1,l_2)}$} be the number of standard tableaux of two lines.
We have

$$f^{(l_1,l_2)} = \frac{l_1-l_2+1}{l_1+1} \left(\begin{array}{c}
l_1+l_2\\ l_1 \end{array} \right)$$


\end{definition}

\begin{proposition}
If \,$l_1 = l_2$ then $$f^{(l_1,l_1)} = \frac{1}{l_1+1} \left(\begin{array}{c}
2l_1\\ l_1 \end{array} \right)$$
which is th n-th Catalan number $C_n$.

\end{proposition}


In general, we can represent a Young tableaux of two (equal) lines as follows :


\begin{figure}[htbp]
\begin{center}
\input{tab_1.pstex_t}
%\caption{Young tableaux of size 2n and height two}
\end{center} \end{figure}

\medskip
\noindent
Then $$2n-k=h$$ will be the second parameter of the tableau.

\medskip
\noindent
\textbf{Example :}

\medskip
\noindent
A Young tableau of size six and height two


\begin{figure}[htbp]
\begin{center}
\input{tab_2.pstex_t}
%\caption{Young tableau of size 6 and height 2}
\end{center} \end{figure}


\medskip
\noindent
Then $$6-4=2$$
we remark that 2 is the last letter of $$\sigma_2(3) =  3 1 2$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dyck Words and Dyck Paths}

Let {\bf w } be a word and {\bf a} a letter, the length of {\bf w} will be denoted by {\bf $|w|$}, and
the number of occurrences of {\bf a} in {\bf w} by {\bf $|w|_a$}.
We denote the empty word by $\epsilon$. If {\bf w = uv }, then {\bf u} is a prefix of {\bf w}.

\begin{definition}
A Dyck word {\bf w} is a word over the alphabet $\sum = \{0,1\}$ with the following
properties :

\begin{itemize}

\item For each prefix of $u$ of $w$, $|u|_1 \geq |u|_0$
\item $|w|_1 = |w|_0$

\end{itemize}

A Dyck path is a path in the first quadrant, which begins at the origin $(0,0)$, ends at
$(2n,0)$. A Dyck path consists {\bf North-east} and {\bf South-east} steps. The number
of Dyck paths of length 2n is the Catalan number
$$C_n = \frac{1}{n+1}  \left(\begin{array}{c}
2n\\ n \end{array} \right)$$
and thus,

$$\sum_{n\geq 0}C_n x^n= \frac{1 - \sqrt{1 - 4x}}{2x}$$


\end{definition}


\noindent
Now, we can refine the number of Dyck words with respect to a parameter like k which is
the number of factors in Dyck words.
The number of Dyck words of length 2n which decompose into $k$ (irreducible Dyck) factors is exactly $l(n,k)$ and their sum (over $k$) equal to the Catalan numbers. For example

$$ a\,a\,a\,a\,b\,b\,b\,b $$

is a Dyck word of length 4 which has one factor.

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

\subsection{Bijections}

In this chapter we will describe the links between some combinatorial
famillies and we try to give certain properties that help us to
understand the connection.



\noindent
$$Codes \leftrightarrow  \sigma_2(n)  \leftrightarrow Young tableaux \leftrightarrow Dyck words $$


\begin{theorem}

Let $\Phi = \{a\} \cup \Phi^+$ be a data structure with a
bi-variate statistics
$$l : \Phi \rightarrow N^2$$
$$s \rightarrow l(s) = (n,k)$$

such that
$$l(\Phi^+) \subset N^+ \times N$$

Suppose that 

We suppose that there exist a function $ d : \Phi^+ \rightarrow \Phi$ 
such that

\begin{enumerate}

\item $d : \Phi_n \rightarrow \Phi_{n-1} $ ($\Phi_n = (pr_1 \circ l)^{-1}(n)$)

\item $ \phi  : \Phi_n \rightarrow \Phi_{n-1} \times  N^+ $
\\ $ s \rightarrow (d(s), k)$ is injective

\item define $\pi=pr_2\circ l$. For all $ s \in \Phi $ we define his code by
$$\chi(s) =(\pi(d^{n-1}(s)),\pi(d^{n-2}(s)),\cdots \pi(d(s)),\pi(s))$$
then $\chi$ is injective.
\end{enumerate}

\end{theorem}


\noindent For example to pass a code $1\,1\,2\,2$ to $\sigma_2(4)$,
$f^{(4,4)}$ and Dyck word of length 8 which decompose into one factor.


%$$
%\begin{figure}
%\begin{tabular}{|c|c|c|}
%\hline
%1\,1\,2\,2&
$$ \sigma_2(2) : 1 \rightarrow 1\,2 \rightarrow 3\,1\,2 \rightarrow 3\,4\,1\,2 $$

$$Dyck \,\, words : a\,b \rightarrow a\,a\,b\,b \rightarrow a\,b\,a\,a\,b\,b \rightarrow a\,a\,b\,a\,a\,b\,b\,b$$

%\hline
%\end{tabular}
%$$


\begin{figure}[htbp]
%\begin{center}
\input{tab_9.pstex_t}
\caption{Young tableau}
%\end{center}
\end{figure}


%\noindent
%In the following table we have the values

%$$
%\begin{figure}
%\begin{tabular}{|c|c|c|c|}
%\hline
%Codes&$\sigma_2(n)$&Dyck's word&Young tableaux\\
%\hline
%1\,1\,\,1\,1& 1\,4\,3\,2& aaaabbbb& 0\\
%\hline
%1\,1\,1\,2& 4\,1\,3\,2& ab$a^3 b^3$& 0\\
%\hline
%1\,1\,2\,1& 3\,4\,2\,1& 0& 0\\
%\hline
%1\,1\,2\,2& 3\,1\,4\,2& 0& 0\\
%\hline
%1\,1\,2\,3& 0& 0& 0\\
%\hline
%1\,2\,1\,1& 0& 0& 0\\
%\hline
%1\,2\,1\,2& 0& 0& 0\\
%\hline
%1\,2\,2\,1& 0& 0& 0\\
%\hline
%1\,2\,2\,2& 0& 0& 0\\
%\hline
%1\,2\,2\,3& 4\,2\,1\,3& 1& ab$a^2 b^2$ab\\
%\hline
%1\,2\,2\,3& 0& 0& 0\\
%\hline
%1\,2\,3\,1& 0& 0& 0\\
%\hline
%1\,2\,3\,2& 0& 0& 0\\
%\hline
%1\,2\,3\,3& 0& 0& 0\\
%\hline
%1\,2\,3\,4& 4\,3\,2\,1& 0& abababab\\
%\hline
%\caption{ JAFF-Duchamp tableau }
%\end{tabular}
%\end{figure}
%$$




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

%\section{Examples and Remarks}






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

\section{Conclusion}

We have presented an toy-model economic behaviour based on local rules and we propose some global function expression which can be also described by three combinatorics structures. By this application, we point out a one-to-one correspondence between three other ballot-like structures. The innovative aspect of this paper deals with a constructive development of the involved bijections.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliographystyle{apalike}
\begin{thebibliography}{99}

%
\bibitem{Ax} B.E. Sagan, {\em The symmetric group}, 1991 .
%
\bibitem{BR} C. Schensted, {\em Longest increasing and decreasing
subsequences}, Canad.J.Math, 13(1961), 179-191.
%
\bibitem{Bo1} M.P. Schützenberger, {\em Quelques remarques sur une construction
de Schensted}, Math. Scand., 12(1963), 117-128.
%
\bibitem{Bo1} Herbert S. Wilf, {\em Ascending subsequences of permutations
and the shape of tableaux}, Journal of Combin. theory, series A 60(1992), 155-157.
%
\bibitem{Bo1} Herbert S. Wilf, {\em The computer-aided discovery of a theorem
 about Young tableaux}, J. Symbolic Computation, series 20 (1995), 731-735.
%
\bibitem{bak} Per Bak, {\em How nature works - the science of self-organized criticality}, Springer Verlag, 1996.
%
\bibitem{schelling} Thomas Schelling, {\em Dynamic models of segregation}, J. of Math. Sociology, vol. 1, 1971.
\end{thebibliography}



\end{document}

