  Probability measures and stochastic dynamics on matrices and on partitions
are related by standard, albeit technical, discrete to continuous scaling
limits. In this paper we provide exact relations, that go in both directions,
between the eigenvalues of the Laguerre process and certain distinguished
dynamics on partitions. This is done by generalizing to the multidimensional
setting recent results of Miclo and Patie on linear one-dimensional diffusions
and birth and death chains. As a corollary, we obtain an exact relation between
the Laguerre and Meixner ensembles. Finally, we explain the deep connections
with the Young bouquet and the z-measures on partitions.
