Modelling a dynamic system of permutations

[vdrn485]

Let us assume a dynamic system which has vector with 'n' components (which are non-negative integers from 1->n) at time t=1. In other words we have a permutation over 1->n at time t=1. Assuming time to be discrete, at any time time 't' , the system evolves such that there are 't' permutations with 'n' components in each. We do not know in advance which permutations in time 't' contributes to the birth of which permutation in time 't+1'. We can assume that each permutation in time 't+1' has a probability distribution over the permutation in time 't' for being its parent.
The only information available is the permutations at each time instance from 1 to T.

What kind of models can be used to represent such a system if we want to find out if the permutations attain a stable state with very less perturbations after certain time period?

 

[Stephen Tashi]

We can assume that each permutation in time 't+1' has a probability distribution over the permutation in time 't' for being its parent.

If the probability distributions are independent of time then you have a Markov process.

You haven't been specific enough to make any particular model a good candidate. Lots of different real life problems match the generalities you presented. For example, pick n stocks and name them 1,2,..n. Let t be the time in days. Let the permutation at time t be this list of stocks ordered from lowest closing price to highest closing price on that day. If there are ties, then break them by giving the stock with the highest market capitalization the higher rank.