18.318 (Spring 2006): Problem Set #5
due May 3, 2006
1. [5] Show that the only r -dierential lattices are direct products of Y s
and Zj s. In particular, the only 1-dierential lattices are Y and Z1 .
2. [5] Let P b e an r -dierential poset. Show that for
18.318 (Spring 2006): Problem Set #2
due March 8, 2006
1. [2+] Let G b e a nite abelian group of order n, written additively.
An n n matrix A = (auv ) over a eld K whose rows and columns
are indexed by G is said to have G-symmetry if there is a function
18.318 (Spring 2006): Problem Set #3
due April 5, 2006
1. A simplicial complex is 2-acyclic (over the eld K ) if is acyclic
and lk(v ) is acyclic for every vertex v of . Here lk denotes link, and
acyclic means that all reduced homology vanishes over K .
18.318 (Spring 2006): Problem Set #4
due April 19, 2006
1. [2] Let Pn b e the set of all planted forests (i.e., graphs for which every
component is a rooted tree) on the vertex set [n]. Let uv be an edge
of a forest F Pn such that u is closer to the root
18.318 (Spring 2006): Problem Set #1
due February 22, 2006
Hand in your best three problems from those below. Reasonable collaboration is permitted, but you should not just copy someone elses solution
or look up a solution from an outside source.
We will
8
A glimpse of Young tableaux.
We dened in Section 6 Youngs lattice Y , the poset of all partitions of all
nonnegative integers, ordered by containment of their Young diagrams.
1111
41
5
1
2111 221 311 32
1111
211 22 31
4
111
3
21
2
11
Circulant Hadamard Matrices
R. Stanley
An n n matrix H is a Hadamard matrix if its entries are 1 and its
rows are orthogonal. Equivalently, its entries are 1 and H H t = nI . In
particular,
det H = nn/2 .
(1)
It is easy to see that if H is an n n Hadamard
4
The Sperner property.
In this section we consider a surprising application of certain adjacency ma
trices to some problems in extremal set theory. An important role will also
b e played by nite groups. In general, extremal set theory is concerned with
n
6
Young diagrams and q -binomial coecients.
A partition of an integer n 0 is a sequence = (1 , 2 , . . .) of integers
i 0 satisfying 1 2 and i1 i = n. Thus all but nitely
many i are equal to 0. Each i > 0 is called a part of . We sometimes
suppress 0s fro
5
Group actions on boolean algebras.
Let us b egin by reviewing some facts from group theory. Suppose that X is
an n-element set and that G is a group. We say that G acts on the set X if
for every element of G we associate a permutation (also denoted ) of