  We introduce a family of polynomials, which arise in three distinct ways: in
the large $N$ expansion of a matrix integral, as a weighted enumeration of
factorisations of permutations, and via the topological recursion. More
explicitly, we interpret the complex Grassmannian $\mathrm{Gr}(M,N)$ as the
space of $N \times N$ idempotent Hermitian matrices of rank $M$ and develop a
Weingarten calculus to integrate products of matrix elements over it. In the
regime of large $N$ and fixed ratio $\frac{M}{N}$, such integrals have
expansions whose coefficients count factorisations of permutations into
monotone sequences of transpositions, with each sequence weighted by a monomial
in $t = 1 - \frac{N}{M}$. This gives rise to the desired polynomials, which
specialise to the monotone Hurwitz numbers when $t = 1$.
  These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join
recursion, a one-point recursion, and the topological recursion. Furthermore,
we conjecture on the basis of overwhelming empirical evidence that the deformed
monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy
remarkable interlacing phenomena.
  An outcome of our work is the viewpoint that the topological recursion can be
used to "topologise" sequences of polynomials, and we claim that the resulting
families of polynomials may possess interesting properties. As a further case
study, we consider a weighted enumeration of dessins d'enfant and conjecture
that the resulting polynomials are also real-rooted and satisfy analogous
interlacing properties.
