  There is a natural map from a symmetric group $S_n$ to a smaller symmetric
group $S_{n-1}$, we write a decomposition of a permutation into a product of
disjoint cycles and remove the element $n$ from this expression. For this
reason there exists the inverse limit $\mathfrak{S}$ of sets $S_n$. We equip
$S_n$ with the uniform distribution (or more generally with an Ewens
distribution) and get a structure of a measure space on $\mathfrak{S}$ (it is
called 'virtual permutations' or 'Chinese restaurant process'), a double
$S_\infty\times S_\infty $ of an infinite symmetric group acts on
$\mathfrak{S}$ by left and right 'multiplications'. We discuss the closure of
$S_\infty\times S_\infty $ in the semigroup of polymorphisms (spreading maps
with spreaded Radon--Nikodym derivatives) of $\mathfrak{S}$. We get formulas
for some polymorphisms, in particular for the center of the closure.
Expressions are sums of multiple convolutions of Dirichlet distributions,
summation sets are certain collections of dessins d'enfant.
