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co.combinatorics – Random permutations with out doble rises (avoiding consecutive sample $underline{123}$) retort

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co.combinatorics – Random permutations with out doble rises (avoiding consecutive sample $underline{123}$)

A permutation avoiding a consecutive sample $underline{123}$ is permutation
$pi = pi_1 pi_2 ldots pi_n$ with the property that there doesn’t exists $i in [1, n-2]$
such that $pi_i < pi_{i+1} < pi_{i+2}$.
occasion: $53241$ is $underline{123}$-avoiding; whereas $314562$ just isn’t $underline{123}$ avoiding, because it accommodates $145$.

Let $a_n$ breathe the variety of $underline{123}$ avoiding permutations,
it’s identified that
$$ frac{a_n}{n!} sim e^{frac{pi}{3sqrt{3}}} left(frac{3sqrt{3}}{{2pi}}capable)^{n+1}.$$

So, the unostentatious rejection sampling is inefficient aircraft for qualify values of $n$.
How does one generate such permutations uniformly at random?

I’m cognizant of Boltzmann sampling.
But possibly you might be cognizant of extra unostentatious and/or sooner algorithms as within the illustration of alternating permutations.

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