We consider the (iterated) Kapranov embedding \(\Omega_n:\overline{M}_{0,n+3} \hookrightarrow \mathbb{P}^1 \times \cdots \times \mathbb{P}^n\), where \(\overline{M}_{0,n+3}\) is the moduli space of stable genus \(0\) curves with \(n+3\) marked points. In 2020, Gillespie, Cavalieri, and Monin gave a recursion satisfied by the multidegrees of \(\Omega_n\) and showed, using two combinatorial insertion algorithms on certain parking functions, that the total degree of \(\Omega_n\) is \((2n-1)!!=(2n-1)\cdot (2n-3) \cdots 5 \cdot 3 \cdot 1\). In this paper, we give a new proof of this fact by enumerating each multidegree by a set of boundary points of \(\overline{M}_{0,n+3}\), via an algorithm on trivalent trees that we call a lazy tournament. The advantages of this new interpretation are twofold: first, these sets project to one another under the forgetting maps used to derive the multidegree recursion. Second, these sets naturally partition the complete set of boundary points on \(\overline{M}_{0,n+2}\), of which there are \((2n-1)!!\), giving an immediate proof of the total degree formula.

Mathematics Subject Classifications: 05E14, 14N10, 05C05, 14H10, 05A19, 05C85

Keywords: Moduli spaces of curves, projective embeddings, multidegrees, trivalent trees

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H_mu(X;q,t)=H_{mu*}(X;t,q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q=0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization t=1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.