Any combinatorical meaning or interpretation of 
$$1^{\alpha\_1}2^{\alpha\_2}3^{\alpha\_3}...s^{\alpha\_s}\alpha\_1!\alpha\_2!...\alpha\_s!$$
for partition $(1^{\alpha\_1},2^{\alpha\_2},3^{\alpha\_3},...,s^{\alpha\_s})\vdash{n}$. 


In addition, this expression is diviser of $n!$. 



Combinatorical meaning of such expression
Given a permutation $(x\_1,\dots,x\_n)$ of numbers from 1 to $n$, we get a new permutation: $x\_1$, $\dots$, $x\_{\alpha\_1}$ are fixed points, $(x\_{\alpha\_1+1},x\_{\alpha\_1+2})$ form a 2-cycle, and so on. Thus we get a permutation with $\alpha\_i$ cycles of length $i$ and each of them is calculated as many times as you ask about.


