  Solutions of the discrete Painlev\'e II hierarchy are shown to be in relation
with a family of Toeplitz determinants describing certain quantities in
multicritical random partitions models, for which the limiting behavior has
been recently considered in the literature. Our proof is based on the
Riemann-Hilbert approach for the orthogonal polynomials on the unit circle
related to the Toeplitz determinants of interest. This technique allows us to
construct a new Lax pair for the discrete Painlev\'e II hierarchy that is then
mapped to the one introduced by Cresswell and Joshi.
