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.
In this paper, we are interested with time series with moderate growth but possibly sudden decay. 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.
Our aim is to describe here a toy-model of the benefit in the following situation.
A capital owner possesses two accounts, say P and R, 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 R. From the account R can be with drawn arbitrary amounts of money and the account must stay positive.
The possible configurations are described the sequences such that In this paper, we build combinatorial structures that allow to modelize and to compute the global behavior of the reserve R by some specific functions. We can consider this result as an emergent function from the basic local rules.
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 [6]. 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 [7]. In such model some local interactionbetween 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
We can define the trajectories of our model by sequences (codes)
(the void sequence) Remark that the preceding table gives the mirror images of the lines of the previous double statistics.
We say that a permutation π of n letters has an increasing subsequences of length k if there are positions
For example 1 2 3 4 5
has increasing subsequences of length 2, at points Let λ be as above. A Young tableau of shape λ , is an array obtained by replacing the squares of the shape of λ by a bijection with the numbers 1,2,…,n.
A tableau T is said to be a standard Young tableau if the rows and columns are increasing sequences. For example below the tableau is standard f , be the number of standard tableaux of two lines. We have
which is th n-th Catalan number n C .
In general, we can represent a Young tableaux of two (equal) lines as follows :
will be the second parameter of the tableau.
Example : A Young tableau of size six and height two
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. For example to pass a code 112 2 to 2 (4) σ , (4 4) f , and Dyck word of length 8 which decompose into one factor.
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 stru
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