Projective embeddings of [formula omitted] and parking functions

Abstract

The moduli space M‾0,n may be embedded into the product of projective spaces P1×P2×⋯×Pn−3, using a combination of the Kapranov map |ψn|:M‾0,n→Pn−3 and the forgetful maps πi:M‾0,i→M‾0,i−1. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height n−3. We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in A2×A3⋯×An−2) is equal to (2(n−3)−1)!!=(2n−7)(2n−9)⋯(5)(3)(1). As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.