Simion’s Type B Associahedron is a Pulling Triangulation of the Legendre Polytope

Ehrenborg, Richard; Hetyei, Gábor; Readdy, Margaret

doi:10.1007/s00454-018-9973-4

Cited by 6 publications

(13 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this reason we call it the cyclohedral triangulation C n . (A related triangulation has been found recently and independently by Ehrenborg, Hetyei and Readdy [19], cf. Section 1.4.…”

Section: Introductionsupporting

confidence: 61%

“…While preparing the final version of this manuscript, we became aware of the recent work by Ehrenborg, Hetyei and Readdy in [19]. There, the authors realize Simion's type B associahedron as a pulling triangulation of the boundary of the Legendre polytope conv{e i − e j : 1 ≤ i, j ≤ n, i = j}, also known as the full root polytope of type A n−1 [19]. We can recover their triangulation by projecting our cyclohedral triangulation of ∆ n ×∆ n along the span of the vectors {(e i , e i ) : i ∈ [n]}.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometry of $\nu $-Tamari lattices in types $A$ and $B$

Ceballos

Padrol

Sarmiento

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of ν-Tamari lattices. In our framework, the main role of "Catalan objects" is played by (I, J)-trees: bipartite trees associated to a pair (I, J) of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path ν = ν(I, J). Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the ν-Tamari lattice introduced by Prévile-Ratelle and Viennot. In particular, we obtain geometric realizations of m-Tamari lattices as polyhedral subdivisions of associahedra induced by an arrangement of tropical hyperplanes, giving a positive answer to an open question of F. Bergeron.The simplicial complex underlying our triangulation endows the ν-Tamari lattice with a full simplicial complex structure. It is a natural generalization of the classical simplicial associahedron, alternative to the rational associahedron of Armstrong, Rhoades and Williams, whose h-vector entries are given by a suitable generalization of the Narayana numbers.Our methods are amenable to cyclic symmetry, which we use to present type B analogues of our constructions. Notably, we define a partial order that generalizes the type B Tamari lattice, introduced independently by Thomas and Reading, along with corresponding geometric realizations. 12 3.1. Flips and the (I, J)-Tamari lattice 132010 Mathematics Subject Classification. 05E45, 05E10, 52B22.

show abstract

“…For this reason we call it the cyclohedral triangulation C n . (A related triangulation has been found recently and independently by Ehrenborg, Hetyei and Readdy [19], cf. Section 1.4.…”

Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Geometry of $\nu $-Tamari lattices in types $A$ and $B$

Ceballos

Padrol

Sarmiento

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…It is called cowell-covered if G is well-covered. For G 1 = G 2 = K d the complete graph on d vertices, the polytope P G 1 + (−P G 2 ) is the Legendre polytope studied by Hetyei et al [Het09,EHR18].…”

Section: Unconditional Reflexive Polytopes and Perfect Graphsmentioning

confidence: 99%

Unconditional Reflexive Polytopes

Kohl

Olsen

Sanyal

2020

Discrete Comput Geom

View full text Add to dashboard Cite

A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets.

show abstract

“…, n + 1} that is both the head and the tail of an arrow; see [14,Lemmas 4.2 and 4.4]. Equivalently, the faces are products of two simplices [11,Lemma 2.2]: we may write them as ∆ I × ∆ J where I, J = ∅, I ∩ J = ∅ and the symbol ∆ K denotes the convex hull of the set {e i : i ∈ K} for K ⊆ {1, 2, . .…”

Section: Preliminariesmentioning

confidence: 99%

“…Triangulations of root polytopes and of products of simplices have been a subject of intense study in recent years [2,5,6,7,13]. Motivated by an observation made in [9], we recently [11] established that the Simion type B associahedron [19] may be realized as a pulling triangulation of the Legendre polytope, defined as the convex hull of all differences of pairs of standard basis vectors in Euclidean space. These vertices can be thought of as arrows between numbered nodes.…”

Section: Introductionmentioning

confidence: 99%

Classification of uniform flag triangulations of the boundary of the full root polytope of type $A$

Ehrenborg¹,

Hetyei²,

Readdy³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

The Legendre polytope is the convex hull of all pairwise differences of the standard basis vectors. It is also known as the full root polytope of type A. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These triangulations fall naturally into three classes: the lex class, the revlex class and the Simion class. We also determine that the refined face counts of these triangulations only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.

show abstract

Simion’s Type B Associahedron is a Pulling Triangulation of the Legendre Polytope

Cited by 6 publications

References 21 publications

Geometry of $\nu $-Tamari lattices in types $A$ and $B$

Geometry of $\nu $-Tamari lattices in types $A$ and $B$

Unconditional Reflexive Polytopes

Classification of uniform flag triangulations of the boundary of the full root polytope of type $A$

Contact Info

Product

Resources

About