Gale-Ryser Theorem

Let $p$ and $q$ be Partitions of a Positive Integer, then there exists a (0,1)-matrix $A$ such that $c(A)=p$, $r(A)=q$ Iff $q$ is dominated by $p^*$.

References

Brualdi, R. and Ryser, H. J. §6.2.4 in Combinatorial Matrix Theory. New York: Cambridge University Press, 1991.

Krause, M. ``A Simple Proof of the Gale-Ryser Theorem.'' Amer. Math. Monthly 103, 335-337, 1996.

Robinson, G. §1.4 in Representation Theory of the Symmetric Group. Toronto, Canada: University of Toronto Press, 1961.

Ryser, H. J. ``The Class ${\mathcal A}({\bf R}, {\bf S})$.'' Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 61-65, 1963.