Скоро Codeforces Round 1133 (Div. 1, Div. 2)! Регистрация здесь.
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In fact there are many generalizations of Cayley theorem (though I didn't know that particular one).
One was given as a problem on Putnam competition one year ago: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3315688&sid=56608f6bc111f1ce8bcfb33ff1220ce3#p3315688
Here is another one I know. Assume that we want to compute number of forests on n vertices with k trees such that vertices with indiced 1, ..., n belong to other trees. Then number of them is equal to knn - k - 1.
Despite the fact that there are many of them I always got problems even with proving basic version :p.