Algebraic Combinatorics (course notes Fall 2008)

129 Pages·2008·1.15 MB·English

by  Geir T. Helleloid

Algebraic Combinatorics
Geir T. Helleloid
Fall 2008
M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
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Contents
1 Enumeration 9
1.1 Lecture 1 (Thursday, August 28): The 12-Fold Way (Stanley [4, Section 1.1]) 9
1.1.1 Stirling Numbers and Bell Numbers . . . . . . . . . . . . . . . . . . . 10
1.1.2 Superclasses (Arias-Castro, Diaconis, Stanley) . . . . . . . . . . . . . 11
1.1.3 Back to the 12-Fold Way . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Lecture 2 (Tuesday, September 2): Generating Functions (Stanley [4, Section
1.1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2 Catalan numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3 q-Analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Lecture 3 (Thursday, September 4): Permutation Enumeration (Stanley [4,
Section 1.1], Wilf [7, Chapter 4] . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.1 Permutation Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.2 Multiset Permutations and q-Analogues . . . . . . . . . . . . . . . . . 23
1.3.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.4 Using Generating Functions to Find Expected Values . . . . . . . . . 25
1.3.5 Unimodality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.6 Cycle Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.7 Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 Lecture 5 (Thursday, September 11): The Exponential Formula (Stanley [5,
Chapter 5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Lecture 6 (Tuesday, September 16): Bijections . . . . . . . . . . . . . . . . . 35
1.6 Lecture 7 (Thursday, September 18): Bijections II . . . . . . . . . . . . . . . 38
1.7 Lecture 8 (Tuesday, September 23): Bijections II (Aigner [1, Section 5.4]) . . 40
1.7.1 The Gessel-Viennot Lemma . . . . . . . . . . . . . . . . . . . . . . . 40
2 Special Topics 43
2.1 Lecture9(Thursday, September25): PermutationPatterns(Bona[2, Chapter
4]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Lecture 10 (Tuesday, September 30): The Matrix Tree Theorem (Stanley [5,
Section 5.6]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.1 Spanning Trees and the Matrix Tree Theorem . . . . . . . . . . . . . 47
2.3 Lecture 11 (Thursday, October 2): The BEST Theorem (Stanley [5, Section
5.6]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.1 The BEST Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
2.3.2 De Bruijn Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Lecture 12 (Tuesday, October 7): Abelian Sandpiles and Chip-Firing Games 52
2.5 Lecture 13 (Thursday, October 9): Mobius Inversion and the Chromatic Poly-
nomial (Stanley [4, Chapter 2]) . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.1 Posets and Mobius Inversion . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.2 Back to the Chromatic Polynomial . . . . . . . . . . . . . . . . . . . 59
2.6 Lecture 14 (Tuesday, October 14): The Chromatic Polynomial and Connections 60
2.6.1 The Graph Minor Theorem . . . . . . . . . . . . . . . . . . . . . . . 61
2.6.2 Hyperplane Arrangements . . . . . . . . . . . . . . . . . . . . . . . . 62
3 The Representation Theory of the Symmetric Group and Symmetric Func-
tions 63
3.1 An Introduction to the Representation Theory of Finite Groups (Sagan [3,
Chapter 1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.1.2 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.3 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Lectures 16 and 17 (Tuesday, October 21 and Thursday, October 23): The
Irreducible Representations of the Symmetric Group (Sagan [3, Chapter 2]) . 66
3.2.1 Constructing the Irreducible Representations (Sagan [3, Section 2.1]) 66
3.2.2 The Specht module Sλ (Sagan [3, Section 2.3]) . . . . . . . . . . . . . 67
3.2.3 The Specht Modules are the Irreducible Modules (Sagan [3, Section 2.4]) 68
3.2.4 Finding a Basis for Sλ (Sagan [3, Section 2.5]) . . . . . . . . . . . . . 70
3.2.5 Decomposition of Mλ (Sagan [3, Section 2.9]) . . . . . . . . . . . . . 71
3.3 Lecture 18 (Tuesday, October 28): The RSK Algorithm (Stanley [5, Section
7.11]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.1 Row Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.2 The Robinson-Schensted-Knuth (RSK) Algorithm . . . . . . . . . . . 73
3.3.3 Growth Diagrams and Symmetries of RSK . . . . . . . . . . . . . . . 74
3.4 Lecture 19 (Thursday, October 30): Increasing and Decreasing Subsequences
(Stanley [5, Appendix A]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Lectures 20 and 21 (Tuesday, November 4 and Thursday, November 6): An
Introduction to Symmetric Functions (Stanley [5, Chapter 7]) . . . . . . . . 78
3.5.1 The Ring of Symmetric Functions . . . . . . . . . . . . . . . . . . . . 78
3.5.2 (Proposed) Bases for the Ring of Symmetric Functions . . . . . . . . 79
3.5.3 Changes of Basis Involving the m . . . . . . . . . . . . . . . . . . . 82
λ
3.5.4 Identities and an Involution . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.5 Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5.6 The Hook Length Formula . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5.7 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
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M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
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M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
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M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
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M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
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Chapter 1
Enumeration
Algebraiccombinatoricsisanew, sprawling, andpoorlydefinedsubjectareainmathematics.
As one might expect, any topic with both an algebraic and a combinatorial flavor can be
called algebraic combinatorics. Topics that are often included in this area that we will not
touch on are finite geometries, polytopes, combinatorial commutative algebra, combinatorial
aspects of algebraic geometry, or matroids. What we will do is start with ten lectures on
the fundamentals of enumerative combinatorics (more or less, the study of counting), includ-
ing methods (generating functions, bijections, inclusion-exclusion, the exponential formula),
standard results (permutation enumeration, enumeration of graphs, identities), and some
special topics (the Marcos-Tardos theorem). We will then spend about four lectures (and
perhaps more toward the end of the semester) on topics in graph theory that have a more
algebraic flavor. This will be followed by about ten lectures on the representation theory
of the symmetric group and symmetric functions. This is the principal topic in the area of
algebraic combinatorics that we will cover, and it will hint at the appearance of enumerative
methods within representation theory and algebraic geometry. The course will finish up with
a few lectures on special topics of particular interest, including the combinatorics of card
shuffling and the enumeration of alternating sign matrices.
1.1 Lecture 1 (Thursday, August 28): The 12-Fold
Way (Stanley [4, Section 1.1])
There are three goals for this lecture. The first is to introduce some of the fundamental
objects in enumerative combinatorics. The second is to foreshadow some of the enumerative
methods that we will discuss in depth in subsequent lectures. The third is to give examples
of why these objects might be of interest outside of combinatorics.
The 12-fold way is a unified way to view some basic counting problems. Let f : N → X
be a function, where |N| = n and |X| = x. It is illustrative to interpret f as an assignment of
n balls to x bins. We arrive at 12 counting problems by placing restrictions on f. On the one
hand, we can count functions f with no restriction, those that are surjective, and those that
areinjective. Ontheotherhand, wecanalsocountfunctionsuptopermutationsofN and/or
permutations of X; an alternative viewpoint is that the balls are either distinguishable or
indistinguishable and the bins are either distinguishable or indistinguishable. We form a
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M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid
chart of the number of distinct functions f under each possible set of restrictions:
N X Any f Injective f Surjective f
Dist Dist (#1)xn (#2)(x) (#3)x!S(n,x)
n
Indist Dist (#4)(cid:0)(cid:0)x(cid:1)(cid:1) (#5)(cid:0)x(cid:1) (#6)(cid:0)(cid:0) x (cid:1)(cid:1)
n n n−x
Dist Indist (#7)S(n,1)+···+S(n,x) (#8)1 if n ≤ x, 0 otherwise (#9)S(n,x)
Indist Indist (#10)p (n)+···+p (n) (#11)1 if n ≤ x, 0 otherwise (#12)p (n)
1 x x
1. (Any f with distinguishable balls and distinguishable bins) Each ball can go in one of
x bins, so there are xn functions.
2. (Injective f with distinguishable balls and distinguishable bins) The first ball can go
into one of x bins, the second can go into one of the x − 1 other bins, and so on, so
there are x(x − 1)···(x − n + 1) functions. This expression occurs often enough to
earn its own notation.
Definition. The falling factorial (x) is defined to be
n
(x) := x(x−1)···(x−n+1).
n
1.1.1 Stirling Numbers and Bell Numbers
3. (Surjectivef withdistinguishableballsanddistinguishablebins)Tochooseasurjective
function f, we have to split the balls up into x groups and pair up each group with a
bin. First, we count the number of ways to split the balls into x groups.
Definition. A set partition of a set N is a collection π = {B ,...,B } of subsets
1 k
called blocks such that the blocks are non-empty, disjoint, and their union is all of N.
For example, the set partitions of {1,2,3} are
{{1},{2},{f (x)f (x)···f (x)3}}(sometimes denoted by 1/2/3)
d1 d2 dm
{{1,2},{3}}(sometimes denoted by 12/3)
{{1,3},{2}}(sometimes denoted by 13/2)
{{1},{2,3}}(sometimes denoted by 1/23)
{{1,2,3}}(sometimes denoted by 123)
The number of set partitions of an n-element set is the Bell number B(n). The number
of set partitions of an n-element set into k blocks is the Stirling number of the second
kind S(n,k). (We will discuss Stirling number of the first kind in a couple lectures.)
Clearly a surjective function f is built by choosing one of the S(n,x) set partitions
of N with x blocks and then assigning blocks to bins in x! ways, so the number of
functions is x!S(n,x).
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