Week 1
☛ Wednesday ☚
Walks in graphs.
Reading: Chapter 1.
Some links that might help with eigenvectors and eigenvalues:
Wiki page
for eigenvalues and eigenvectors,
Wiki page for
diagonalization,
HMC
tutorial,
Calculating
eigenvalues and eigenvectors,
Diagonalizing
a matrix.
☛ Friday ☚
Walks on the complete graph.
Reading: Chapter 1.
Week 2
☛ Monday ☚
Random walks.
Homework due Monday
☛ Wednesday ☚
Posets, chains, antichains, Sperner property.
Homework due Wednesday
- Reading: Chapter 4 through the definition of the Sperner property.
- Turn in:
- Let \(G\) be the graph with vertices
\(v_1,v_2,v_3,v_4\), obtained from the complete graph by deleting the
edge \(\{v_1,v_3\}\). Use the corollaries to Theorem 1.1 to
- Give a formula for the number of closed walks of length \(\ell\).
- Describe these closed walks for the each of the cases \(\ell=0,1,2,3\).
Hint: to help factor the characteristic polynomial, note that both \(0\) and
\(-1\) are roots.
- Show that the trace of a square matrix is the sum of its eigenvalues.
(Hint: use the Jordan canonical form and standard properties of the
trace.)
- Chapter 1, exercise 2. You'll need Lemma 1.7 and the trick we used to
find the eigenvalues for the adjacency matrix for the complete graph from
the matrix of all 1s.
- Chapter 1, exercise 4.
Week
2 HW solutions
☛ Friday ☚
Order matchings from order-raising operators.
Homework due Friday
- Reading: Chapter 4 through the proof of Lemma 4.5.
Week 3
☛ Monday ☚
\(B_n\) has the Sperner property.
Homework due Monday
- Reading: Finish Chapter 4.
☛ Wednesday ☚
The symmetric group. Group actions.
Homework due Wednesday
Week
3 HW solutions
☛ Friday ☚
The quotient poset \(B_n/G\).
Homework due Friday
- Reading: Chapter 5 through Theorem 5.8.
Week 4
☛ Monday ☚
Review homework. Sperner property for \(B_n/H\).
Homework due Monday
☛ Wednesday ☚
Sperner property for \(B_n/H\). Young diagrams.
Homework due Wednesday
Week
4 HW solutions
☛ Friday ☚
The combinatorics of juggling patterns.
Homework due Friday
Week 5
☛ Monday ☚
\(q\)-binomial coefficients and Young diagrams. Enumeration under symmetry.
Homework due Monday
- Reading: Chapter 6 and beginning of Chapter 7.
☛ Wednesday ☚
Enumeration under symmetry. Polya counting.
Homework due Wednesday
Week
5 HW solutions
☛ Friday ☚
Polya counting.
Homework due Friday
Week 6
☛ Monday ☚
Proof of Polya's theorem.
Homework due Monday
☛ Wednesday ☚
Generating functions: first example; the ring of formal power series.
Homework due Wednesday
Week
6 HW solutions
☛ Friday ☚
Calculus of ordinary generating functions.
Homework due Friday
- Reading: Generatingfunctionology, Chapters 1 and 2.