Summary: Packing Ferrers Shapes
February 22, 2002
Answering a question of Wilf, we show that if n is sufficiently large, then one cannot cover an
n × p(n) rectangle using each of the p(n) distinct Ferrers shapes of size n exactly once. Moreover,
the maximum number of pairwise distinct, non-overlapping Ferrers shapes that can be packed in
such a rectangle is only (p(n)/ log n).
A partition p of a positive integer n is an array p = (x1, x2, · · · , xk) of positive integers so that
x1 x2 · · · xk and n = k
i=1 xi. The xi are called the parts of p. The total number of distinct
partitions of n is denoted by p(n). A Ferrers shape of a partition p = (x1, x2, · · · , xk) is a set of n
square boxes with sides parallel to the coordinate axes so that in the ith row we have xi boxes and all
rows start at the same vertical line. The Ferrers shape of the partition p = (4, 2, 1) is shown in Figure
1. Clearly, there is an obvious bijection between partitions of n and Ferrers shapes of size n.
If we reflect a Ferrers shape of a partition p with respect to its main diagonal, we get another
shape, representing the conjugate partition of p. Thus, in our example, the conjugate of (4,2,1) is