  $\tau$-functions of certain Painlev\'e equations (PVI,PV,PIII) can be
expressed as a Fredholm determinant. Further, the minor expansion of these
determinants provide an interesting connection to Random partitions. This paper
is a step towards understanding whether the $\tau$-function of Painlev\'e II
has a Fredholm determinant representation. The Ablowitz-Segur family of
solutions are special one parameter solutions of Painlev\'e II and the
corresponding $\tau$-function is known to be the Fredholm determinant of the
Airy Kernel. We develop a formalism for open contour in parallel to the one
formulated in \cite{CGL} in terms of the Widom constant and verify that the
Widom constant for Ablowitz-Segur family of solutions is indeed the determinant
of the Airy Kernel. Finally, we construct a suitable basis and obtain the minor
expansion of the Ablowitz-Segur $\tau$-function.
