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 correspon- dence, and link it with a Hurwitz partition function and a Hodge partition by operators in a G L(∞) group. Then, from a W1+∞ constraint of the partition function of r-spin inter- section numbers, we get a W1+∞ constraint for the Hodge partition function. The W1+∞ constraint completely determines the Schur polynomials expansion of the Hodge partition function.