Math 249 Problem Set 1
Problems from Stanley. Notation X [Y ] means Exercise X in the second edition, corresponding to Y in the rst (see Appendix First Edition Numbering in the second edition).
1.8 [1
Math 249 Problem Set 3
Problems from Stanley (Volume 2).
5.11, 5.13(a,b), 5.20, 5.23
Additional problems.
1. Prove that the Eulerian polynomials An (x) satisfy the the following more symmetrical
recur
Math 249 Problem Set 2
Problems from Stanley (Vol. 1, second edition).
1.5, 1.22(b), 1.48(b), 1.51, 1.76, 1.102
Additional problems.
1. By expanding the right hand side in partial fractions, show dire
Math 249 Problem Set 4
From Stanley (Vol. I 2nd ed): Exercise 3.66
Additional problems.
1. In class I sketched out some of this problem and the next. Now I ask you to ll in the
details, and compute so
Generating function for connected
graphs
Calculating Z conn x , q
n ,k
g conn n , k x n q k , where g conn n , k is the
number of isomorphism classes of connected simple graphs with n vertices
and k e
Math 249 Problem Set 5
This problem set is due Friday, May 15. You can turn it in either by e-mail or at my
oce, 855 Evans Hall - slip it under the door if you come by when I am not there.
1. Recall f
MATH 249 PROBLEM SET 5 (DUE THURSDAY APR. 27)
(1) Prove (using only the denition of representations) that the symmetric group Sn ,
n 2, has exactly two one-dimensional representations: the trivial rep
MATH 249 PROBLEM SET 1 (DUE SEPTEMBER 19)
(1) Let P be a nite poset, and m N. Let m be the chain poset 1 < 2 < < m.
Show that the following numbers are equal:
(a) The number of surjective order-preser
MATH 249 PROBLEM SET 4 (DUE APRIL 1)
(1) Show that the two denitions of matroid that we saw in class one in terms of independent sets and one in terms of circuits are cryptomorphic. In other words, sh
MATH 249: SUGGESTIONS FOR FINAL PROJECTS
(1) Bruhat order of Coxeter groups and shellability:
Anders Bjorner, Michelle Wachs, Advances in Math., 43, 1982.
(2) Posets, regular CW complexes and Bruhat o