Limit distributions of the number of vertices of given degree in the forest of a random mapping with a given number of cycles

We consider a random one-to-one mapping of a set of size n into itself whose graph contains m cycles. Upon deleting the cyclic vertices and arcs, we obtain the subgraph called the forest of the mapping. We find limit distributions of the number of vertices of a given degree in the forest as n -> infinity and m = O(ln n). A one-to-one mapping S of the finite set T-n = {1, 2, . . . , n} into itself can be represented in the form of the matrix s = ( 1 2 . . . n ) s(1) s(2) . . . s(n)