# 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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