Problem 15E. Show with a Ferrers diagram that the number of
partitions of n + k into k: parts equals the number of partitions of
n into at most k: parts. (This is (15.2).)
Many theorems about partitions can be proved easily by repre-
senting each partition by a diagram of dots, known as a Ferrers
diagram. Here we represent each term of the partition by a row
of dots, the terms in descending order, with the largest at the top.
Sometimes it is more convenient to use squares instead of dots (in
this case the diagram is called a Young diagram by some authors
but we shall use Ferrers diagram as the name for the figure). For
example, the partition (5,4,2,1) of 12 is represented by each of the
diagrams of Fig. 15.1.
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