Home Syllabus Schedule Links Homework     Homework Starred* problems should be completed if you are enrolled in the course at the 6000 level. I recommend (but do not require) that all students read and attempt these problems. If you make some significant progress, you should turn it in. Homework 10 11, due Dec 2 Bóna p339: 23, 27, 28 Bóna p476: 25, 27 Homework 10, due Nov 17 at 1pm Bóna p176: 27, 28, 32, 33, 40 A. Recall that the moment generating function of a random variable X is the exponential generating function for the sequence E(X^{n}). Show that if X has the moment generating function e^{t2/2}, then X^{2} has the moment generating function (1  2t)^{1/2}.
Note: e^{t2/2} is the mgf for the standard normal distribution. (1  2t)^{1/2} is the mgf for the chisquared (1) distribution. Modulo the fact that an mgf determines the distribution of its random variable, Problem A proves that the square of a standard normal random variable is chisquared. Homework 9, due Nov 3 at 1pm A. If Δ is a simplicial complex, then the fvector of Δ is the sequence f_{i}^{Δ} = #{faces of Δ having i vertices} Find the generating function for f_{i}^{Δ} in the case where Δ is a single dsimplex. B. If Δ and Γ are simplicial complexes, then the join of Δ and Γ is the simplicial complex Δ * Γ with vertex set V(Δ) ∪ V(Γ), and all faces of the form δ ∪ γ for a face δ of Δ and a face γ of Γ. Using the Product Lemma and/or MetaTheorem, relate the generating functions for the fvectors of Δ, Γ, and Δ * Γ. C. Find a power series representation for sqrt(1  16x^{2}), and relate it to the Catalan numbers. D. If c_{n} is the nth Catalan number, prove the identity (Sum from i = 0 to n of c_{2i}c_{2n  2i}) = 4^{n} c_{n} Hint: Use generating functions. The preceding problem and the Product Lemma are both related. E. Show that the number of ways to divide an ngon into triangles by adding (noncrossing) edges between existing vertices is counted by a Catalan nuber. Homework 8, due Oct 27 at 1pm Bóna p176: 25, 26, 48, 49a, 49b A. Show that the Catalan numbers count the number of NE lattice paths that do not pass above the diagonal line y = x.
Homework 7, due Oct 20 at 1pm Bóna p143: 20, 22, 23, 26, 29* (see also problem 9), 32 Bóna p176: 24, 47
Homework 6, due Oct 13 at 1pm Bóna p126: 34, 36, 40, 43, 52, 55 A. Find an arrangement of "hyperplanes" (lines) in R^{2} such that the lines are in 11 correspondence with elements of order 2, and the regions are in 11 correspondence with elements of the dihedral group D_{2n}. B*. Describe all subspaces of R^{4} that are preserved by the action of S_{4} by permutation matrices. (We described at least one such subspace in class!)
Homework 5, due Oct 6 at 1pm Bóna p126: 31, 32, 33, 38, 39, 41, 42
Homework 4, due Sep 19 at 1pm Bóna p107: 21, 28, 30*, 31, 32, 33, 34, 36 Hint on 31: n^{2} + 2n = (n + 1)n + n. Hint on 30: It might be helpful to show that there are a lot of partitions having all parts small.
Homework 3, due Sep 12 at 1pm Bóna p79: 33, 38 Bóna p107: 17, 24, 26, 27, 35
Homework 2, due Sep 8 at 1pm Bóna p53: 49, 53 Bóna p79: 28, 36, 40, 43, 45
Homework 1, due Aug 29 at 1pm Bóna p53: 26, 28, 34, 35, 39, 48 A. Write a detailed proof of Theorem 3.5 from Bóna.
