Overview
Research indicates that the counts of flats in a matroid, when ordered by rank, do not necessarily form a unimodal sequence. This finding explicitly disproves Rota's conjecture concerning the unimodality of such sequences. The study further demonstrates that these sequences can exhibit an arbitrary number of peaks.
Research Context
The investigation directly addresses Rota's conjecture, which posited that the sequence of flat counts in a matroid, organized by rank, would be unimodal. Unimodality, in this context, refers to a sequence that first increases monotonically and then decreases monotonically. The research provides a counterexample to this long-standing conjecture.
Approach
The construction employed to disprove Rota's conjecture initiates with identifying a generalized theta graph. This specific type of graph was selected because it exhibits severe failure of log-concavity, a property often related to unimodality. Log-concavity failure in this initial step is a critical element in the subsequent construction.
Following the identification of the generalized theta graph, the researchers utilized direct sums. This technique was applied to introduce multiple instances of log-concavity failure within the sequence. By aggregating these failures, the method aimed to create a sequence with multiple turning points.
The final step in the construction involved applying Whittle's $q$-lift. This specific mathematical operation was used to transform the properties established from the direct sums into a structure that produces a matroid. The resulting matroid's flat counts then demonstrated the presence of numerous peaks, thereby serving as a direct counterexample to the unimodality conjecture.
Findings
- Rota's conjecture, stating that matroid flat counts by rank form a unimodal sequence, is disproven.
- The sequence of flat counts in a matroid can possess an arbitrary number of peaks.
- The construction utilized a generalized theta graph exhibiting severe log-concavity failure.
- Direct sums were used to introduce multiple points of log-concavity failure.
- Whittle's $q$-lift construction produced a matroid with flat counts demonstrating many peaks.