  We prove that Fredholm determinants of the form det(1-K_s), where K_s is the
restriction of either the discrete Bessel kernel or the discrete {}_2F_1 kernel
to {s,s+1,...}, can be expressed through solutions of discrete Painleve II and
V equations, respectively.
  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 exp(\eta(u+1/u)) and (1+u)^z(1+\xi/u)^{z'}.
  The proofs are based on a general formalism involving discrete integrable
operators and discrete Riemann-Hilbert problem. A continuous version of the
formalism has been worked out in math-ph/0111007.
