C&O 430/630, Fall 2011
Assignment 2
Due Wednesday, November 2, in class.
1. (a) Let F be the species of endofunctions: FX is the set of all functions : X X, and
f = f f 1 for any bijection f : X Y . By viewing an endofunction as a directed
graph, nd a nat

C&O 430/630, Fall 2010
Assignment 4
Due Wednesday, December 15, NOON.
1. We say that a skew partition / is a ribbon, if there is at most one path between any two
boxes with horizontal and vertical steps; hence / does not contain a 2 2 square of boxes.
The

C&O 430/630, Fall 2010
Assignment 2
Due Friday, October 29, in class.
1. (a) Let F be the species of endofunctions: FX is the set of all functions : X X, and
f = f f 1 for any bijection f : X Y . By viewing an endofunction as a directed
graph, find a natu

C&O 430/630, Fall 2010
Assignment 1
Due Friday, October 8, in class.
1. (a) Let A(x) =
n
n0 an x
P
C[x]. Let be a primitive mth root of unity. Show that
m1
X
1 X
A( k x) =
amn xmn .
m
n0
k=0
(b) Show that for ` = 0, 1, 2,
X n 2n
2n
1<
<
+ 1.
3
3k + `
3
k

Lecture Notes for C&O 430/630
Fall 2010
Kevin Purbhoo1
September 11, 2010
1 based
in part on course materials written by I.P. Goulden and D.G. Wagner
Chapter 1
Formal power series
In this chapter, we develop the theory of formal power series and formal La

Lecture Notes for C&O 430/630
Fall 2010
Kevin Purbhoo1
October 4, 2010
1 based
in part on course materials written by I.P. Goulden and D.G. Wagner
Chapter 1
Formal power series
In this chapter, we develop the theory of formal power series and formal Laure

C&O 430/630, Fall 2011
Assignment 3
Due Monday, November 21, in class.
1. A tournament is an orientation of the complete graph. Let Tn be the set of all tournaments
on vertices cfw_1, . . . , n. For Tn , let wtj ( ) be the out-degree of vertex j, and let

C&O 430/630, Fall 2011
Assignment 4
Due Monday, December 5, in class.
1. Column insertion is dened analogously to row insertion, with the roles of rows and columns
reversed. Given T SSYT() and a number a, we dene a T to be the tableau obtained
by column-m

C&O 430/630, Fall 2011
Assignment 1
Due Friday, October 14, in class.
1. Use Hensels Lemma to prove the following.
(a) Suppose that A(x), B(x) Q[x], are formal power series such that A(0) = B(0) = 1
and A(x)B(x) = B(x)A(x) . Prove that A(x) = B(x).
(b) Le

C&O 430/630, Fall 2010
Assignment 3
Due Monday, November 22, in class.
1. Prove that the crystal reflection operators Ri on SSYT(/) satisfy the braid relations:
Ri Ri+1 Ri = Ri+1 Ri Ri+1 .
(Hint: First use sliding to reduce to the case where i = 1 and the