The following material can be found in a number of sources, including Sections
7.3 7.5, 7.7, 7.10 7.11, 7.15 16 of Stanleys Enumerative Combinatorics Volume
2.
1 Elementary and Homogeneous Symmetric Func
tions
A polynomial in n variables, P (x1 , x2 , .
Todays lecture notes cover the Oriented Matrix Theorem, which is discussed in
Sections 9 and 10 of Richard Stanleys Topics in Algebraic Combinatorics lecture
notes. The proof presented in class more closely resembles the bijective proof in
Section 4.4 of
The following is an outline of the material covered April 6th and 8th in class.
This material can be found in Chapter 5 of Stanleys Enumerative Combinatorics
Volume 2. Proofs of most of the results are in class notes.
1
Exponential Generating Functions
De
The following material can be found in several sources including Sections 14.9
14.13 of Algebraic Graph Theory by Chris Godsil and Gordon Royle, as well as
the papers Chip-ring Games on Graphs by Anders Bjrner , Lsl Lovsz, and
o
a o
a
Peter Shor (1991), a
1
Introduction to Partitions
A partition of n is an ordered set of positive integers = (1 , 2 , . . . , n ) such that
i i = n and 1 2 k .
Let P (n) denote the set of all partitions of n with p(n) = |P (n)| and p(0) = 1. For
example, the partitions of 4 ar
The material for this lecture can be found in several sources, for example see
Section 4.1 of William Fultons book Young Tableaux.
1
Proof of Schensteds Theorem
Theorem (Schensted). Let be a permutation of cfw_1, 2, . . . , n written in one-line
notation.
1
Mbius Function on Posets
o
This material closely follows selections from Chapter 3 of Enumerative Combina
torics 1 by Richard Stanley.
1.1
New Posets from Old
If P and Q are posets on disjoint sets, then the disjoint union (or direct sum) of
P and Q is
1
Perfect Matchings and Domino Tilings
Deniton. A Perfect Matching of a graph G = (V, E) is a set M E of
distringuished edges such that each vertex v V is incident to exactly one edge of
M.
Notice that in particular that this implies that |M | = |V |/2 an
1
Recurrence Relations and Generating Functions
Given an innite sequence of numbers, a generating function is a compact way of
expressing this data. We begin with the notion of ordinary generating functions.
To illustrate this denition, we start with the
1
Partially Ordered Sets II: Dilworths Theorem
Denition. An anitchain A of a poset P is a subset of elements of P such that
for all x, y A, x y and y x.
We denote the levels of a graded poset P as Pi where Pi = cfw_x P : rank(x) = i.
Remark. Observe that