Research

Partially ordered sets, or posets, are ubiquitous in mathematics and many other fields.  They capture the relationships between elements of the set when not every pair of elements is directly comparable.  In this paper, we initiate the study of interval-closed sets of finite posets from enumerative and dynamical algebraic perspectives. In particular, we define the action rowmotion on interval-closed sets as a product of toggles from the generalized toggle group. Our main theorem is an intricate global characterization of rowmotion on interval-closed sets, which we show is equivalent to the toggling definition. We also study specific posets; we enumerate interval-closed sets of ordinal sums of antichains, completely describe their rowmotion orbits, and prove a homomesy result involving the signed cardinality statistic. Finally, we study interval-closed sets of product of chains posets, proving further results about enumeration and homomesy. 

Lying at the intersection of graph theory and voting theory, a k-majority tournament is a direct graph whose vertices are the set of candidates or choices given to voters. An edge is drawn from candidate u to candidate v if and only if they are preferred by at least k of the voters, where k is some set value greater than half the size of the electorate.  In a weighted k-majority tournament, each edge is labeled with the number of voters preferring u to v.  In this paper we define a measure (the approval gap) for how strongly a dominating set in such a tournament beats the next most popular candidate.  We prove upper and lower bounds on the approval gap, and construct tournaments having a specified approval gap within a given range dependent on k.   

In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer code rotation, the reverse, the complement, the Foata bijection, and the Kreweras complement. The statistics studied relate to familiar notions such as inversions, descents, and permutation patterns, and also more obscure constructs. Besides the many new homomesy results, we discuss our research method, in which we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies.