  We prove the equivalence of ensembles for Bernoulli measures on $\mathbb{Z}$
conditioned on two conserved quantities under the situation that one of them is
spatially inhomogeneous. For the proof, we extend the classical local limit
theorem for a sum of Bernoulli independent sequences to those multiplied by
linearly growing weights. The motivation comes from the study of random Young
diagrams. We discuss the relation between our result and the so-called Vershik
curve which appears in a scaling limit for height functions of two-dimensional
Young diagrams.
