Permutations with Cycles
This is a digital sketch inspired by the equality:
where 
denotes the number of permutations of, say, ![[n]](http://neeldhara.com/wp-content/plugins/latex/cache/tex_de504dafb2a07922de5e25813d0aaafd.gif)
with 
cycles. The equality is nearly self-explanatory — you simply knock off one of the elements (let’s do away with 
) and stare at permutations of ![[n-1]](http://neeldhara.com/wp-content/plugins/latex/cache/tex_933031fa29f65eadbed6b08024c5dbb7.gif)
.
Click here for a full explanation »
In particular, look at permutations of that have 
cycles: such permutations can be extended to permutations of with cycles by simply adding the element and letting the permutation take to itself. On the other hand, for all permutations that have cycles, we might “insert”, so to speak, the element into one of the existing cycles. A moments reflection tells us that this can be done in one of 
ways, for there are as many slots for insertion — when considered in total.
These observations written out amount to the identity we had.
