Projective embeddings of (M)over-bar0,n and parking functions

The moduli space (M) over bar (0,)n may be embedded into the product of projective spaces P-1 xP(2) x center dot center dot center dot xP(n-3), using a combination of the Kapranov map vertical bar psi(n vertical bar) : (M) over bar (0,)n -> Pn-3 and the forgetful maps pi(i): (M) over bar (0,i) -> (M) over bar (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 A(2) x A(3) center dot center dot center dot x A(n-2)) is equal to (2( n - 3) - 1)!! = (2n - 7)(2n - 9) center dot center dot center dot (5)(3)(1). As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial. (C) 2021 Elsevier Inc. All rights reserved.