  We derive an asymptotic formula for $A(n,j,r)$ the number of integer
partitions of $n$ into at most $j$ parts each part $\le r$. We assume $j$ and
$r$ are near their mean values. We also investigate the second largest part,
the number of parts $\ge 2$, etc. We show that the fraction of the partitions
of an even integer $n$ that are graphical, ie. whose parts form the degree
sequence of a simple graph, is $O(\ln^{-1/2} n)$. Probabilistic results are
used in our discussion of graphical partitions. The George Szekeres circle
method is essential for our asymptotic results on partitions. We determine the
distributions defined by the successive ranks of partitions, generalizing the
result of Erdos and Richmond for the rank of a partition.
