The Cayley Trick and Triangulations of Products of Simplices

By Francisco Santos. 2003

82 citations recorded by the source corpus.

We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and non-regular triangulations ofl × � k grow, respectively, as k �(k) and 2 (k 2 ) . For the special case of � 2 × � k , we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangu- lations up to k = 15, show that the number grows as ek 2 /2+o(k

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/faded9071f920affbc3e47ad11dffd3e0d9d12d1

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In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Electrical and Electronic Engineering

Citations in the corpus

Tropical convexity (2003, background): “It is worth noting that via the Cayley Trick [19], Theorem 1 is equivalent to saying that tropical complexes generated by r points in TPn−1 are in bijection with the regular mixed subdivisions of the dilated simplex r∆n−1.”
On the rank of a tropical matrix (2003, methodology): “See [22] for more details.”
Tropical hyperplane arrangements and oriented matroids (2007, methodology): “This is not true in general [18], and there are only a few polytopes whose triangulations are known to be ip-connected, including convex polygons, cyclic polytopes [15], and products n 1 2 [ 17 ].”
Permutohedra, Associahedra, and Beyond (2005, methodology): “Santos [30] used this trick to study triangulations of the product of two simplices.”
Flag arrangements and triangulations of products of simplices (2006, background): “(This bijection is only a special case of the more general Cayley trick, which is discussed in detail in [38].”