Combinatorics Through Guided Discovery, 2004a

C.6. Supplementary Problems 167
6. Find the EGF for the number of graphs on n vertices in which every vertex has
degree 2.
7. Recall that a cycle of a permutation cannot be empty.
(a) What is the generating function for the number of cycles on an even number
of elements (i.e. permutations of an even number n of elements that form
an n-cycle)? Your answer should not have a summation sign in it. Hint: If
y = ∑ ∞
i=0 x2i
2i
, what is the derivative of y?
(b) What is the generating function for the number of permutations on n elements
that are a product of even cycles?
(c) What is the generating function for the number of cycles on an odd number
of elements?
(d) What is the generating function for the number of permutations on n elements
that are a product of odd cycles?
(e) How do the generating functions in parts b and d of this problem related to
the generating function for all permutations on n elements?