Math 146: Algebraic Combinatorics
Homework 5
This problem set is due Friday, April 30. Do problem *3.1 and 3.21, in addition to my problems.
A remark on where we are in the book: We’re sortof in sections 2.3 and 2.4, but we also did sections 3.14
and 3.15, and I will say some things about partitions as covered in 3.16.
A hint on problem 3.1, which is quite difficult otherwise: Given a partition on
n
into odd parts, you can
center its Ferrers diagram. Then you can divide the centered diagram into L shapes, which is then a partition
of the same
n
into distinct parts. For example, given 18
=
7
+
5
+
3
+
3, the centered diagram is
to yield 18
=
7
+
6
+
4
+
1. You should describe more rigorously what the L shapes do, and prove that it is
a bijection between the two types of partitions.
Update: I decided to make problem 3.1 extra credit.
GK5.1
The partition number
p
(
n
)
in section 3.16 can be described as the number of ways to express
n
cents with unordered coins, with one coin of each denomination 1 cent, 2 cents, 3 cents, etc. Suppose
instead that the coins are ordered, or that we are looking at ordered partitions where 3
=
2
+
1 is
different from 3
=
1
+
2. Let’s call the number of such partitions
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 Spring '10
 GregKuperberg
 Fibonacci number, Generating function, Integer sequences, Ferrers diagram

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