Vol. 11 No 1 - Pi Mu Epsilon

·'
38
Official IIME Tee Shirts
Get Your's Today!
Problem. A painter is high atop a 40 foot ladder leaning against a 60 foot house
at an angle such that the ratio of the lengths of the legs of the right triangle formed
is the golden section. If the base of the ladder is moving away from the house at
at the speed of light divided by Avagadro's number, what are the thoughts of the
mathematician watching the ensuing disaster?
j
v = dh
dt
dv _ g
dt -
F = mzmeGR
11ll 3
0 r> =FxR
b 1
= ----:---
a 1+1+11
Ft = J.tkN
Solution. She hopes that none of the paint will get on her new IIME Tee Shirt.
TIME Journal, Vol. 11, No. 1, pp 39-44, 1999. 39
PARTITIONS AND YOUNG'S LATTICES
DAVID WARREN*
An earlier paper in this journal [2] explored the relationships between partitions
and their Young lattices. The purpose of this article is to expand that work. We
begin with a review of partitions, lattices, and Hasse diagrams.
A partition of a natural number N is a finite sequence of natural numbers n1, n2,
... , nm in non-increasing order such that l::i=l mni = N. We will represent partitions
by Ferrer's diagrams as shown in the first example. See [1] for details.
Example 1. Ferrer's diagram for the partition (4, 2, 1) of seven is shown in Figure
1.
FIG. 1.
A lattice is a partially ordered set that has the property that any two elements
x and y have a least upper bound (join) and a greatest lower bound (meet). If we
order the set of all partitions of a set P by containment, the resulting poset is a lattice
called Young's lattice. We will represent the resulting lattices as a Hasse diagram.
The bottom element in a Hasse diagram is assigned a rank of one. The rank of
a subpartition in Young's lattice is the number of squares in Ferrer's diagram of that
partition.
In [2] it was shown that Young's lattice for the partition (n) is a single chain as
shown in Figure 2, and Young's lattice for the partition (n, 2) (with n > 2) is shown
IIME Tee shirts are white, Haines, BEEFY-T, pre-shrunk 100% cotton. The front
has a large IIME shield. The back of the shirt is decorated with the colorful IIME
tesselation of the plane designed by Doris Schattschneider, in the IIME colors of gold,
lavender and violet. Shirts are available in large and X-large. The price is only $10
per shirt, which includes postage and handling.
To obtain a shirt, send your check or money order, payable to IIME, to:
Rick Poss,
Mathematics- IIME
St. Norbert College
100 Grant St.
DePere, WI 54115
FIG. 2.
in Figure 3.
*Hendrix College
n
D
(1)