Download Math 462 Homework 3: Problems on Combinatorics and Graph Theory - Prof. Ioana Dumitriu and more Assignments Mathematics in PDF only on Docsity!
Math 462 HOMEWORK # 3 due Friday, January 25
Reading Assignment: Read chapters 8.2.1, 8.2.2, 9.1, 9.2 (9.1 and 9.2 were studied last quarter; presumably you can just skim through and recall the important results).
Written Assignment: There are 4 mandatory problems in this assignment, make sure you do them all. Problem 5 is optional (it earns “brownie points”, which have no explicit value, but may put me in a particularly good mood when setting your grade).
Problem 1. Let Pn be the number of ways to place n open parentheses and n closed parentheses in a list in an “allowable” way (such that, among the first k parentheses, there are never more closed ones than open ones, for each 1 ≤ k ≤ 2 n). E.g., for n = 4, ((())()) is allowable, but ())(()() is not. Show that Pn = (^) n^1 +
( 2 n n
(the Catalan numbers) by constructing a bijection between allowable lists of n closed and n open parantheses and Dyck paths of length 2n.
Problem 2. Let there be 2n points on a circle. Pair up all points and draw the chords corresponding to each pair; we call the resulting picture an n-configuration. Let Tn be the number of n-configurations in which no two chords intersect. For example, for n = 2, Tn = 2, as you can see below.
1
2
3
4
1
2
3
4
pairing (1,4) and (2,3) pairing (1,2) and (3,4)
There is one more possible 2-configuration for the chords, but the two chords intersects in this case:
1
2
3
4
pairing (1,3) and (2,4)
Show that Tn = (^) n^1 +
( 2 n n
in one of two ways:
- either by showing that Tn satisfies the recurrence Tn+1 =
∑n k=0 TkTn−k, and that T 0 = 1, T 1 = 1.
- or by finding a bijection between configurations of n non-intersecting chords and either Dyck paths of length 2n or “allowable” ways of placing n closed and n open parentheses in a row (the latter are the objects of Problem 1).
Problem 3. This problem is based on Problem 19 from Chapter 6 (page 169).
(a) Given r(n) as the number of n-permutations whose square is the identity, we have proved in Homework 2 that r(n + 2) = r(n + 1) + (n + 1)r(n) , with r(0) = r(1) = 1. We will now find the exponential generating function R(x) for r(n). The solution to Problem 19, given on page 180, is unnecessarily complicated (do not look at it, lest you be lead astray). Multiplying the recurrence by xn+1/(n + 1)! and summing over all n ≥ 0, find a first-order differential equation satisfied by R(x) and R′(x). Solve it (like we did for the Bell numbers, in class). b) Given a prime p, let r(n) now be the number of n-permutations whose pth power is the identity. We have proved in Homework 2 that r(n + p) = r(n + p − 1) + (n + p − 1)(n + p − 2) · · · (n + 1)r(n) , with r(0) = r(1) = · · · = r(p − 1). Again, let R(x) be the exponential generating function for r(n). Adapt the calculation from part a) to find a first-order differential equation satisfied by R(x) and R′(x). Solve it (note that it should agree with what you found in part a), for p =2).
Problem 4. This problem is based on Problem 42 from Chapter 9 (page 201). Let G be a simple graph on vertex set [n] in which each vertex has degree two. a) Prove that G is a union of disjoint cycles. b) Let g(n) be the number of graphs described above, and set g(0) = 1, g(1) = 0, g(2) = 0. Find a recurrence for g(n).
Problem 5 (optional; “brownie points” problem). Find an infinite set of numbers n such that the complete graph Kn can be decomposed into edge-disjoint Hamiltonian cycles. For these numbers, indicate how to construct the decomposition.
