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A decorated tree approach to random permutations in substitution-closed classes. (English) Zbl 07225521
This paper analyzes random permutations from substitution-closed classes via a probabilistic approach. Given a substitution-closed class $$C$$ with the set $$S$$ of simple permutations for $$i$$ in $$[n]$$, the generating function of $$S$$ is also denoted by $$S$$ for convenience. Its radius of convergence is denoted by $$\rho_S$$. For a permutation $$v$$ and a pattern $$\pi$$, denote by $$c$$-$$occ(\pi,v)$$ the number of consecutive occurrences of pattern $$\pi$$ in $$v$$. Suppose that $$S'(\rho_S)\ge1/(1+\rho_S)^2-1$$. Consider a uniform random permutation $$v_n$$ of size $$n$$ in $$C$$, where $$n$$ is any positive integer. By identifying the packed forest associated with a uniform random permutation in a substitution-closed class as a conditioned mono-type Galton-Waston forest, it is shown that for each pattern $$\pi\in C$$, there exists $$\gamma\in[0,1]$$ such that $$\frac1n c$$-$$occ(\pi,v_n)\rightarrow\gamma$$ in probability as $$n$$ tends to infinity.

##### MSC:
 60C05 Combinatorial probability 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 05A05 Permutations, words, matrices
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