  We give a new proof of the cut-and-join equation for the monotone Hurwitz
numbers, derived first by Goulden, Guay-Paquet, and Novak. Our proof in
particular uses a combinatorial technique developed by Han.
  The main interest in this particular equation is its close relation to the
quadratic loop equation in the theory of spectral curve topological recursion,
and we recall this motivation giving a new proof of the topological recursion
for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.
