Random Walk vs Branching process

1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs with probability 1/N!. The probability $Pm(N)$ that the inversion number equals $m$ for a random permutation is well known as the Mahonian distribution in probability theory. The statistic is applied to some problems like mixing of diffusing particles. Here is just an example http://arxiv.org/abs/1010.2563

2) On the other hand,it is well known that simple ranadom walk is an example of a Markov chain. What I am looking for is to understand how the MAhonian distribution can be represented in terms of a branching process. In other words I am looking for a branching process which will represents the Mahonian distribution: the basic statistical characteristics of the inversion number including the average, the variance, and more generally, the probability distribution function.

Do you think that the representation is possible? Any ideas on the topic?

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