2020/12/18 by Nicholas J. Williams, Williams, Nicholas J.
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Primary: 06A07 #Secondary: 05B45
paper · pdf · doi:10.48550/arxiv.2012.10371
openalex publication_date 2020/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff--Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff--Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff--Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.