Given an integer partition of n , we consider the impartial combinatorial game LCTR in which moves consist of removing either the left column or top row of its Young diagram. We show that for both normal and misère play, the optimal strategy can consist mostly of mirroring the opponent’s moves. We also establish that both LCTR and Downright are domestic as well as returnable, and on the other hand neither tame nor forced. For both games, those structural observations allow for computing the Sprague–Grundy value any position in O ( log ( n ) ) time, assuming that the time unit allows for reading an integer, or performing a basic arithmetic operation. This improves on the previously known bound of O ( n ) due to Ilić (2019). We also cover some other complexity measures of both games, such as state–space complexity, and number of leaves and nodes in the corresponding game tree .

We define a Dowling lattice generalization of the k-equal partition lattice and the h,k-equal signed partition lattice. We use the theory of lexicographical shellability to show that it has the homotopy type of a wedge of spheres, to describe its Betti numbers, and to give a basis for its homology in terms of labelled trees. For cyclic groups, the h,k-equal Dowling lattice arises as the lattice of intersections of a complex subspace arrangement. We use Whitney homology to compute the cohomology of the complement of this arrangement. Our results generalize and unify previous research.