Triangulations of cyclic polytopes and higher Bruhat orders

Abstract

Recently Edelman and Reiner suggested two poset structures, scriptS1(n, d) and scriptS2(n, d) on the set of all triangulations of the cyclic d‐polytope C(n, d) with n vertices. Both posets are generalizations of the well‐studied Tamari lattice. While scriptS2(n, d) is bounded by definition, the same is not obvious for scriptS1(n, d). In the paper by Edelman and Reiner the bounds of scriptS2(n, d) were also confirmed for scriptS1(n, d) whenever d≤5, leaving the general case as a conjecture.

The Cayley trick and triangulations of products of simplices

“…T [2] T [3] T [4] T [5] T [6] T [7] T [8] T [12] T [11] T [10] T [9] T [13] T [14] T [15] T [16] T [20] T [19] T [18] T [17] T In the first inequality we assume that D ≤ N/2, which always happens for N = kle 2 l and D = (k − 1)(l − 1); in the last inequality we have used our assumption that D = (k − 1)(l − 1) ≥ 6.…”

“…On the one hand, there are not many examples where the graph of flips is known to be connected. Essentially, only the case of dimension at most 2 (classical), codimension at most 3 [1] and that of cyclic polytopes [14]. On the other hand, since all triangulations of products of simplices are unimodular, the graph of flips has a very direct interpretation in toric algebraic geometry; see Theorem 2 below.…”

Algebraic and Geometric Methods in Enumerative Combinatorics

“…The triangulations of the cyclic polytope are unusually well behaved; see [DLRS10,Ram97] and the references therein. In particular, when m = d + 4, this is one of the few polytopes whose triangulations have been enumerated exactly (and non-trivially) [AS02].…”

Problems and Results around the Erdös-Szekeres Convex Polygon Theorem

“…These arguments however cannot be extended to even dimensions: it is known [42] that every triangulation of a cyclic d-polytope, d even, consists of the same number of simplices.…”