We prove that Fredholm determinants of the form $\det(1-K\sb s)$,
where $K\sb s$ is the restriction of either the discrete Bessel kernel
or the discrete $\sb 2F\sb 1$-kernel to $\{s, s + 1,\ldots\}$, can be
expressed, respectively, through solutions of discrete Painlevé
II (dPII) and Painlevé V (dPV) equations.
¶ These Fredholm determinants can also be viewed as distribution
functions of the first part of the random partitions distributed
according to a Poissonized Plancherel measure and a $z$-measure, or as
normalized Toeplitz determinants with symbols $e\sp
{\eta(\zeta+\zeta\sp {-1})}$ and $(1 +\sqrt {\xi}\zeta)\sp z(1 +\sqrt
{\xi}/\zeta)\sp {z\sp \prime}$.
¶ The proofs are based on a general formalism involving discrete
integrable operators and discrete Riemann-Hilbert problems. A
continuous version of the formalism has been worked out in [BD].