Undergraduate Seminar in Discrete Mathematics — 18.304
Prof. Olivier Bernardi
April 4, 2011
QUIZ 2
NAME: .....................................................
Question 1 (Alex’s talk).
On the graph below, the order on the edges of the graph is 1 < 2 < 3 < 4 < 5 < 6 < 7.
4
3
6
5
1
2
7
a) What are the active edges for this spanning tree?
b) What is the contribution of this spanning tree to the Tutte polynomial TG (x, y) (for the spanning
tree expansion of the Tutte polynomial)?
Question 2 (Samantha’s talk).
We consider the binary tree of rational numbers constructed using Stern’s diatomic sequence.
a) What are the children of 13/7 in the binary tree of the rationals?
b) What is the 64th term a64 in Stern’s diatomic sequence? (remember that a0 = 0 and a1 = 1).
2
Question 3 (Michael’s talk).
a) What is Cayley’s Formula for the Number of Trees with n vertices?
b) Draw all the trees for n = 3.
Question 4 (Robert’s talk).

0.5 0.2


We consider a Markov chain with states {1, 2, 3} and transition matrix M =  0.3 0.4

0.4 0.1
For instance the probability to go from state 1 to state 3 in one step is 0.3. We suppose
0.3



0.3  .

0.5
that at
time 0 the probability of being in state 1 is 0.5 and the probability of being in state 2 is 0.5 (the
probability to be in state 3 is 0). What is the probability of being in state 3 at time 2?
3
Question 5 (Adriana’s talk).
We consider a source with letters {a, b, c, d, e, f } having respective frequencies F (a) = 5, F (b) = 2,
F (c) = 10, F (d) = 7, F (e) = 3, F (f ) = 1. Draw a Huffman Tree that corresponds to the following
source, and give the code for each of the letters.
Question 6 (Nan’s talk).
We consider the problem of finding to the minimum of x1 − x2 under the constraints −2x1 + x2 ≥ 3,
3x1 − x2 = 7, and x1 ≥ 0. Give the standard form of this linear program problem.
4
Question 7 (Yifan’s talk).
Suppose that you want to hire a professor and that you have n candidates. You decide to adopt
the following strategy: you will interview but not hire the r first candidates, and then hire the next
candidate which is better than all the candidates before her/him.
a) What is the probability that you will hire the candidate r+3 and that this was the best candidate?
b) What would be a good choice of r if there are n = 2718 candidates (and you have a lot of time)?
Question 8 (Karen’s talk).
Let tn be the number of rooted plane trees with n vertices having either 0, 2 or 4 children. Let
P
n
T (x) = ∞
n=0 tn x be the generating function.
a) Find an equation of the form T (x) = x f (T (x)).
b) What does the Lagrange inversion formula say about the coefficient tn = [xn ]T (x)?
5
Question 9 (Haitao’s talk).
We consider the weighted graph below.
4
3
v
6
5
1
2
7
a) Draw the minimum spanning tree.
b) In which order will the edges be added using Prim’s algorithm with starting vertex v?
Question 10 (Andrew’s talk).
a) Find a closed form for the number f λ of Standard Young Tableaux for the partition λ = (n, n).
Do you recognize this number?
b) Suppose that a partition µ is contained in another partition ν and that they have the same
number of Standard Young Tableaux. What can you say about the shape of ν?