  The generalized Kontsevich model (GKM) is a one-matrix model with arbitrary
potential. Its partition function belongs to the KP hierarchy. When the
potential is monomial, it is an $r$-reduced tau-function that governs the
$r$-spin intersection numbers. In this paper, we present an ordered exponential
representation of monomial GKM in terms of the $W_{1+\infty}$ operators that
preserves the KP integrability. In fact, this representation is naturally the
solution of a $W_{1+\infty}$ constraint that uniquely determines the
tau-function. Furthermore, we show that, for the cases of Kontsevich-Witten and
generalized BGW tau-functions, their $W_{1+\infty}$ representations can be
reduced to their cut-and-join representations under the reduction of the even
time independence and Virasoro constraints.
