  Generalized Kontsevich Matrix Model (GKMM) with a certain given potential is
the partition function of $r$-spin intersection numbers. We represent this GKMM
in terms of fermions and expand it in terms of the Schur polynomials by
boson-fermion correspondence, and link it with a Hurwitz partition function and
a Hodge partition by operators in a $\widehat{GL}(\infty)$ group. Then, from a
$W_{1+\infty}$ constraint of the partition function of $r$-spin intersection
numbers, we get a $W_{1+\infty}$ constraint for the Hodge partition function.
The $W_{1+\infty}$ constraint completely determines the Schur polynomials
expansion of the Hodge partition function.
