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{\bf Fr\'ed\'erique Bassino, Mathilde Bouvel and Dominique Rossin}
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{\bf Enumeration of Pin-Permutations}
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In this paper, we study the class of pin-permutations, that is to say
of permutations having a pin representation. This class has been
recently introduced by Brignall, Huczynska and Vatter who used it to
find properties (algebraicity of the generating function, decidability
of membership) of classes of permutations, depending on the simple
permutations this class contains. We give a recursive
characterization of the substitution decomposition trees of
pin-permutations, which allows us to compute the generating function
of this class, and consequently to prove, as it is conjectured by
Brignall, Ru{\v{s}}kuc and Vatter, the rationality of this generating
function. Moreover, we show that the basis of the pin-permutation
class is infinite.
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