On an alternating sum of factorials and Stirling numbers of the first kind: trees, lattices, and games

We study an alternating sum involving factorials and Stirling numbers of the first kind. We give an exponential generating function for these numbers and show they are nonnegative and enumerate the number of increasing trees on $n$ vertices that are won by the second of two players when interpreted as a game tree. We also give a simple description of the quotient from the weak order to the Tamari lattice in terms of plane trees, and give bijections between plane trees, 213-avoiding permutations, and 312-avoiding permutations. Finally, for a rooted tree, we give equivalent characterizations of when it describes a game won by the first or second player in terms of the rank-generating function of the lattice of prunings and the Euler characteristic of an associated real variety.