Dylan Heuer, Ph.D.
Instructor
- Milwaukee WI UNITED STATES
- Mathematics
Dr. Dylan Heuer's research area is in combinatorics and deals with various generalizations of permutations and alternating sign matrices.
Education, Licensure and Certification
Ph.D.
Mathematics
North Dakota State University
2021
M.S.
Mathematics
North Dakota State University
2018
B.A.
Mathematics and Music
Concordia College
2013
Biography
Areas of Expertise
Accomplishments
NDSU Mathematics Department Graduate Student Teaching Award
2019
Event and Speaking Appearances
Partial Permutohedra
Algebra & Discrete Mathematics Seminar North Dakota State University
2021-03-23
Selected Publications
Partial permutation and alternating sign matrix polytopes
SIAM Journal on Discrete MathematicsHeuer, D.; Striker, J.
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice.
Chained permutations and alternating sign matrices—Inspired by three-person chess
Discrete MathematicsHeuer D.; Morrow, C.; Noteboom, B. ; S. Solhjem; Striker, J.; Vorland, C.
We define and enumerate two new two-parameter permutation families, namely, placements of a maximum number of non-attacking rooks on chained-together chessboards, in either a circular or linear configuration. The linear case with corresponds to standard permutations of , and the circular case with and corresponds to a three-person chessboard.
Partial Alternating Sign Matrix Bijections and Dynamics
Electronic Journal of CombinatoricsHeuer, D.
2024-04-05
We investigate analogues of alternating sign matrices, called partial alternating sign matrices. We prove bijections between these matrices and several other combinatorial objects. We use an analogue of Wieland's gyration on fully-packed loops, which we relate to the study of toggles and order ideals. Finally, we show that rowmotion on order ideals of a certain poset and gyration on partial fully-packed loop configurations are in equivariant bijection.
On nu Faces of Partial Alternating Sign Matrix Polytopes
Electronic Journal of CombinatoricsHeuer, D.; Solhjem, S.; Striker, J.
We define and study the (ν/λ)-partial alternating sign matrix polytope, motivated by connections to the Chan-Robbins-Yuen polytope and the ν-Tamari lattice. We determine the inequality description and show this polytope is a face of the partial alternating sign matrix polytope. We show that the (ν/λ)-partial ASM polytope is an order polytope and a flow polytope.
Partial Alternating Sign Matrix Bijections and Dynamics
The Electronic Journal of CombinatoricsDylan Heuer
2024-04-05
We investigate analogues of alternating sign matrices, called partial alternating sign matrices. We prove bijections between these matrices and several other combinatorial objects. We use an analogue of Wieland's gyration on fully-packed loops, which we relate to the study of toggles and order ideals. Finally, we show that rowmotion on order ideals of a specific poset and gyration on partial fully-packed loop configurations have the same orbit structure.
