Geodesic
On the semigroup of transformations of a finite set generated by random generators
Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 99-110
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We consider the semigroup generated by random mappings and random bijective mappings of a finite set $\Omega_n$ of cardinality $n$ into itself. We study the question when this semigroup includes all mappings of $\Omega_n$ into itself with a fixed cardinality $k$ of the image of the set $\Omega_n$. As $n\to\infty$, the ranges of $k$ are given where this inclusion holds with probability tending to zero or one, and two domains of values of $k$ where the inclusion holds with intermediate probability. 
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M. V. Fedyukin. On the semigroup of transformations of a finite set generated by random generators. Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 99-110. http://geodesic.mathdoc.fr/item/DM_2001_13_2_a4/

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