Counterexamples to conjectures about Subset Takeaway and counting linear extensions of a Boolean lattice

We develop an algorithm for efficiently computing recursively defined functions on posets. We illustrate this algorithm by disproving conjectures about the game Subset Takeaway (Chomp on a hypercube) and computing the number of linear extensions of the lattice of a 7-cube and related lattices.

💡 Research Summary

This paper presents the development and application of an efficient algorithm for computing recursively defined functions on partially ordered sets (posets). The authors demonstrate the power of this algorithm by solving two distinct combinatorial problems: disproving longstanding conjectures about the impartial game “Subset Takeaway” and computing the exact number of linear extensions for the 7-dimensional Boolean lattice for the first time.

The first major focus is the game of Subset Takeaway, which is a variant of Chomp played on the Boolean lattice B_n of all subsets of an n-element set. A move consists of choosing a non-empty subset and removing it and all its supersets. The player unable to move loses. While a strategy-stealing argument proves the first player has a winning advantage for n>0, the concrete winning move was unknown. Gale and Neyman (1982) conjectured that taking the largest element (the full n-set) is always a winning move and that in the poset P_{n,k} (subsets of size at most k), the first player loses if and only if (k+1) divides n. Using their computational algorithm, the authors disprove both conjectures. They show that for n=7, the only winning opening move in P_{7,7} (i.e., B_7) is to take a subset of size 4, not the full set. Furthermore, they compute that the first player loses in P_{7,3} but wins in P_{7,6}, providing a counterexample to the divisibility conjecture. They also tabulate Grundy values for small P_{n,k}, revealing that a simple pattern where the Grundy value equals n modulo (k+1) does not hold. The paper describes a reduction technique using involutions (symmetries of order 2) to deduce outcomes for some larger posets from smaller ones.

The second major application is counting linear extensions of the Boolean lattice. A linear extension is a total ordering compatible with the subset partial order. The number e(B_n) was known only up to n=6. The authors’ algorithm recursively sums the number of linear extensions of subposets obtained by removing maximal elements. Leveraging the same symmetry-reduction and canonicalization techniques developed for the game analysis, they achieve a massive computational speedup. They successfully calculate e(B_7), a number approximately 6.3 * 10^137, marking the first time this value has been determined. The computation took under 2.5 hours, compared to 16 hours reported for the previous state-of-the-art calculation of e(B_6). As further demonstrations, they derive and prove a closed-form formula for e(P_{n,2}) and provide computed values for e(P_{n,3}) up to n=7.

A significant portion of the paper is dedicated to explaining the computational methodology. The main challenge is the explosive growth of the state space, which is essentially the number of antichains (Dedekind numbers). The authors’ key innovation is to exploit the symmetric group action to consider game positions only up to isomorphism. They define a canonical form for a downwards-closed set (a game position) by using an invariant vector (for each element, count the number of j-sets containing it) and applying permutations to bring this invariant and the set itself into a lexicographically minimal form. This canonicalization, combined with memoization (storing results in a hash table), makes the exhaustive search feasible for n=7. The paper concludes with a table of sample running times and memory usage for different computations on P_{7,3} and P_{7,7}, providing a practical benchmark for the efficiency of their approach.

In summary, this work makes substantial contributions on two fronts: it settles open conjectures in combinatorial game theory regarding Subset Takeaway, and it achieves a breakthrough in the exact enumeration of linear extensions for a fundamental combinatorial structure. Furthermore, it introduces a general and highly efficient algorithmic framework for poset-based recursive computations, with potential applicability to a wide range of other problems in order theory and combinatorics.