Combinatorics Through Guided Discovery, 2004a

64 3. Distribution Problems
Problem 160. Explain the relationship between partitions of k into n parts
and lists x 1 , x 2 ,..., x n of positive integers that add to k with x 1 ≥ x 2 ≥
... ≥ x n . Such a representation of a partition is called a decreasing list
representation of the partition.
◦ Problem 161. Describe the relationship between partitions of k and lists or
vectors (x 1 , x 2 ,...,x n ) such that x 1 +2x 2 +...kx k = k. Such a representation
of a partition is called a type vector representation of a partition, and it is
typical to leave the trailing zeros out of such a representation; for example
(2, 1) stands for the same partition as (2, 1, 0, 0). What is the decreasing list
representation for this partition, and what number does it partition?
Problem 162. How does the number of partitions of k relate to the number
of partitions of k +1whose smallest part is one? (h)
When we write a partition as λ = λ 1 ,λ 2 ,...,λ n , it is customary to write the list
of λ i s as a decreasing list. When we have a type vector (t 1 , t 2 ,...,t m ) for a partition,
we write either λ =1 t 1
2 t2 ···m t m
or λ = m t m
(m − 1) t m−1
···2 t 2
1 t 1
. Henceforth we
will use the second of these. When we write λ = λ i 1
1
λ i 2
2 ···λ i n
n , we will assume that
λ i >λ i+1 .
3.3.3 Ferrers and Young Diagrams and the conjugate of a partition
The decreasing list representation of partitions leads us to a handy way to visualize
partitions. Given a decreasing list (λ 1 ,λ 2 ,...λ n ), we draw a figure made up of
rows of dots that has λ 1 equally spaced dots in the first row, λ 2 equally spaced dots
in the second row, starting out right below the beginning of the first row and so on.
Equivalently, instead of dots, we may use identical squares, drawn so that a square
touches each one to its immediate right or immediately below it along an edge.
See Figure 3.3.1 for examples. The figure we draw with dots is called the Ferrers
diagram of the partition; sometimes the figure with squares is also called a Ferrers
diagram; sometimes it is called a Young diagram. At this stage it is irrelevant
which name we choose and which kind of figure we draw; in more advanced work
the squares are handy because we can put things like numbers or variables into
them. From now on we will use squares and call the diagrams Young diagrams.
Figure 3.3.1: The Ferrers and Young diagrams of the partition (5,3,3,2)