All orders asymptotic expansion of large partitions

The generating function which counts partitions with the Plancherel measure (and its q-deformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the Gromov-Witten invariants of the Xp = O(p− 2)⊕O(−p) → P Calabi-Yau 3-fold, and we prove a conjecture of M. Mariño, that the generating functions Fg of Gromov–Witten invariants of Xp, come from a matrix model, and are the symplectic invariants of the mirror spectral curve.