Packing Ferrers Shapes

Abstract

Answering a question of Wilf, we show that if $n$ is sufficiently large, then one cannot cover an $n\times p(n)$ 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 $\Theta (p(n)/\log n).$

7 pages, 4 figures