Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse matrices (inverse LUE). Their large-$N$ expansions are generating functions double (strictly weakly) monotone Hurwitz numbers which count constrained factorisations in symmetric group. The two can be compared combined with a duality relation proved \[F. D. Cunden, F. Mezzadri, N. O’Connell, J. Simm, Moments random hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) combinatorial proof reflection formula between moments LUE at genus zero and, ii) new functional strictly numbers. main result resolves integrality conjecture formulated P. Vivo, Correlators Wigner–Smith time-delay matrix chaotic cavities, A 49 (2016), 18, 18LT01, 20 pp] on quantum transport. precise description given here may cast light concordance semiclassical theories.