Homology of Newtonian Coalgebras

Ehrenborg, Richard; Readdy, Margaret

doi:10.1006/eujc.2002.0620

Cited by 9 publications

(7 citation statements)

References 14 publications

(13 reference statements)

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“…There are proofs which have both used and revealed the underlying algebraic structure. See for instance [15,22]. When the ab-index Ψ(P ) is written in terms of c and d, the resulting polynomial is called the cd-index.…”

Section: The Cd-indexmentioning

confidence: 99%

Affine and Toric Hyperplane Arrangements

Ehrenborg

Readdy

Slone

2009

Discrete Comput Geom

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Section: The Cd-indexmentioning

confidence: 99%

Affine and Toric Hyperplane Arrangements

Ehrenborg

Readdy

Slone

2009

Discrete Comput Geom

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“…This is Exercise 69c in [12]. Also this lemma is implicit in the two papers [3,7]. A three-line proof is as follows.…”

Section: Eulerian Binomial Posetsmentioning

confidence: 89%

Classification of the factorial functions of Eulerian binomial and Sheffer posets

Ehrenborg

Readdy²

2007

Journal of Combinatorial Theory, Series A

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We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2 n−1 , the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n) = n! has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function B(n) = 2 n−1 as the doubling of an upside-down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra B X or the infinite cubical lattice C <∞ X . We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.

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“…There is an extensive literature on these algebras, and this survey will only touch the surface. For a deeper look, the reader is directed to (in chronological order) [24,34,15,8,1,36,13,9,44,54,21].…”

Section: Algebrasmentioning

confidence: 99%

The $cd$-Index: A Survey

Bayer

2019

Preprint

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This is a survey of the cd-index of Eulerian partially ordered sets. The cd-index is an encoding of the numbers of chains, specified by ranks, in the poset. It is the most efficient such encoding, incorporating all the affine relations on the flag numbers of Eulerian posets. Eulerian posets include the face posets of regular CW spheres (in particular, of convex polytopes), intervals in the Bruhat order on Coxeter groups, and the lattices of regions of oriented matroids. The paper discusses inequalities on the cd-index, connections with other combinatorial parameters, computation, and algebraic approaches.

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Homology of Newtonian Coalgebras

Cited by 9 publications

References 14 publications

Affine and Toric Hyperplane Arrangements

Affine and Toric Hyperplane Arrangements

Classification of the factorial functions of Eulerian binomial and Sheffer posets

The $cd$-Index: A Survey

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