Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
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72 3. Distribution Problems<br />
⇒<br />
12. Let c(k, n) be the number of ways for k children to hold hands to form n circles,<br />
where one child clasping his or her hands together and holding them out to form a<br />
circle is considered a circle. Find a recurrence for c(k, n). Is the family of numbers<br />
c(k, n) related to any of the other families of numbers we have studied? If so, how?<br />
⇒ 13. How many labeled trees on n vertices have exactly four vertices of degree 1?<br />
⇒<br />
14. The degree sequence of a graph is a list of the degrees of the vertices in nonincreasing<br />
order. For example the degree sequence of the first graph in Figure 2.3.3<br />
is (4, 3, 2, 2, 1). For a graph with vertices labeled 1 through n, theordered degree<br />
sequence of the graph is the sequence d 1 , d 2 ,...,d n in which d i is the degree of<br />
vertex i. For example the ordered degree sequence of the first graph in Figure 2.3.1<br />
is (1, 2, 3, 3, 1, 1, 2, 1).<br />
(a) How many labeled trees are there on n vertices with ordered degree sequence<br />
d 1 , d 2 ,...,d n ?<br />
(b) How many labeled trees are there on n vertices with with the degree sequence<br />
in which the degree d appears i d times?
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