  We examine the two-dimensional U(N) Yang-Mills theory by using the technique
of random partitions. We show that the large N limit of the partition function
of the 2D Yang-Mills theory on S^2 reproduces the instanton counting of 4D N=2
supersymmetric gauge theories introduced by Nekrasov. We also discuss that we
can take the ``double scaling limit'' by fixing the product of the N and cell
size in Young diagrams, and the effective action given by Douglas and Kazakov
is naturally obtained by taking this limit. We give an interpretation for our
result from the view point of the superstring theory by considering a brane
configuration that realizes 4D N=2 supersymmetric gauge theories.
