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 [1.4 ], 1.19(a) [1.8(a) ], 1.32 [1.12 ], 1.46(a) [1.17(a)
Math 249 Problem Set 3
Problems from Stanley (Volume 2).
5.11, 5.13(a,b), 5.20, 5.23
1. Prove that the Eulerian polynomials An (x) satisfy the the following more symmetrical
recurrence than the one in the proof of Stanley, Prop. 1.4.4
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
1. By expanding the right hand side in partial fractions, show directly that the generating
function for Stirling numbers
Math 249 Problem Set 4
From Stanley (Vol. I 2nd ed): Exercise 3.66
1. In class I sketched out some of this problem and the next. Now I ask you to ll in the
details, and compute some terms by hand of generating functions we calculated
Generating function for connected
Calculating Z conn x , q
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 edges.
General: compat :
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 from class that the Schur functions s (x1 , . . . , xn )
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 representation
and the sign representation.
(2) Let G be a
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-preserving maps : P m.
(b) The number of chains = I0 < I1 < <
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, show
that if M is a matroid according to the independent
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 order:
Anders Bjorner, Posets, regular CW complexes and