We study piecewise-linear and birational lifts of Schützenberger promotion, evac-uation, and the RSK correspondence defined in terms of toggles. Using this perspective, we prove that certain chain statistics in rectangles shift predictably under the action of these maps. We then use this to construct piecewise-linear and birational versions of Rubey’s bijections between fillings of equivalent moon polyominoes that preserve these chain statis-tics, and we show that these maps form a commuting diagram. We also discuss how these results imply Ehrhart equivalence and Ehrhart quasi-polynomial period collapse of certain analogues of chain polytopes for moon polyominoes.

We use the octahedron recurrence to give a simplified statement and proof of a formula for iterated birational rowmotion on a product of two chains, first described by Musiker and Roby. Using this, we show that weights of certain chains in rectangles shift in a predictable way under the action of rowmotion. We then define generalized Stanley-Thomas words whose cyclic rotation uniquely determines birational rowmotion on the product of two chains. We also discuss the relationship between rowmotion and birational RSK and give a birational analogue of Greene's theorem in this setting.

In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical and computational statistics. These developments have found applications in a growing number of areas, including biology, neuroscience, economics, and social sciences. Naturally, new connections continue to be made with other areas of mathematics and statistics. We outline three such connections: to statistical models used in educational testing, to a classification problem for a family of nonparametric regression models, and to phase transition phenomena under uniform sampling of contingency tables. We illustrate the motivating problems, each of which is for algebraic statistics a new direction, and demonstrate an enhancement of related methodologies.

We define a birational map between labelings of a rectangular poset and its associated trapezoidal poset. This map tropicalizes to a bijection between the plane partitions of these posets of fixed height, giving a new bijective proof of a result by Proctor. We also show that this map is equivariant with respect to birational rowmotion, resolving a conjecture of Williams and implying that birational rowmotion on trapezoidal posets has finite order.

Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper, we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and provide direction for future work.