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The combinatorial community is awesome! On 6 Jan 2021, me and Jean-Luc Baril, we submitted to arXiv a paper about a [transformation à la Foata for special kinds of descents and excedances in permutattions](https://arxiv.org/abs/2103.13092). At the end of this paper we present two conjectures that straighten our result. On March 24, 2021, Bin Han, Jianxi Mao, Jiang Zeng anounced the proof of our second conjecture. The paper above presents their work. The first conjecture remains open.
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I wonder if we can obtain asymptotic results in the same manner...
A related [question](https://mathoverflow.net/questions/384683/random-permutations-without-double-rises-avoiding-consecutive-pattern-underli/384947#384947) on Mathoverflow.
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The article provides an extension of Boltzmann sampling to the realm of labeled combinatorial structures, enumerated by exponential generating functions and constructed using the box product. ---- [Boltzmann sampling](https://en.wikipedia.org/wiki/Boltzmann_sampler) is a general, pretty fast, technique for generate uniformely random combinatorial structures specified by structural equations. The key idea of Boltzmann sampling can be illustrated by the following example. Suppose you have a generating function $$f(x) = x + x^2 + x^2 + x^3 + x^3 + x^3 + x^3 + x^4 ...$$ corresponding to the one object of size one, two objects of size 2, four objects of size 3, etc. If you set, for example, $x = \frac{1}{2}$ then $$f\left(\frac{1}{2}\right) = \frac{1}{2} + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^4 ...$$ Now let's take $x$ such that $f(x) < \infty$, and chose an integer $k$. As you think, it quickly becomes clear that you can use this to chose a random object of size $k$. The probability equals precisely $\frac{x^k}{f(x)}$. For a given $k$ it is uniform. The elaboration of this idea in the context of structural equations (a.k.a. recursive decompositions, [symbolic method](https://en.wikipedia.org/wiki/Symbolic_method_%28combinatorics%29)) leads you to Boltzmann sampling technique. And knowing the values of generating functions together with corresponding recursive decompositions will allow you to generate a random object according to a uniform probability measure for any size. Shifting $x$ you will shift the mean size of a generating object. See [Boltzmann Samplers For The Random Generation Of Combinatorial Structures](https://www.researchgate.net/publication/2562585_Boltzmann_Samplers_For_The_Random_Generation_Of_Combinatorial_Structures) paper by Duchon, Flajolet, Louchard and Schaeffer for such elaboration.
See also these 2 related questions on MathOverflow: https://mathoverflow.net/questions/384683/random-permutations-without-double-rises https://mathoverflow.net/questions/14863/random-alternating-permutations
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