On Posets and Hopf Algebras

Ehrenborg, Richard

doi:10.1006/aima.1996.0026

Cited by 163 publications

(213 citation statements)

References 7 publications

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“…Likewise, we get the graded risk polynomial from R(G; f ) by replacing each f g by a |g| . We note that the graded risk polynomial is related to Ehrenborg's quasi-symmetric function encoding [12] of the flag f -vector of the chain complex of G ′ .…”

Section: Discussionmentioning

confidence: 99%

Evolution on distributive lattices

Beerenwinkel

Eriksson

Sturmfels

2006

Journal of Theoretical Biology

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Abstract. We consider the directed evolution of a population after an intervention that has significantly altered the underlying fitness landscape. We model the space of genotypes as a distributive lattice; the fitness landscape is a real-valued function on that lattice. The risk of escape from intervention, i.e., the probability that the population develops an escape mutant before extinction, is encoded in the risk polynomial. Tools from algebraic combinatorics are applied to compute the risk polynomial in terms of the fitness landscape. In an application to the development of drug resistance in HIV, we study the risk of viral escape from treatment with the protease inhibitors ritonavir and indinavir.

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Section: Discussionmentioning

confidence: 99%

Evolution on distributive lattices

Beerenwinkel

Eriksson

Sturmfels

2006

Journal of Theoretical Biology

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“…They also show up in real hyperplane arrangements, Prüfer codes, quasisymmetric functions, and Hopf algebras, but detailing all of these connections would lead us too far afield. (see [1], [4], [5], [6], [8], [24], [46], [47], or [45] for details.) We conclude with one final illustration: the associahedron.…”

Section: Discussionmentioning

confidence: 99%

Noncrossing Partitions in Surprising Locations

McCammond

2006

The American Mathematical Monthly

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“…For general properties of F (Q), see [21] and [6]. In particular, we have the following Proposition 1.8.…”

Section: Quasisymmetric Functions and Poset Enumerationmentioning

confidence: 92%

“…We investigate some of the algebraic properties of the map [u, v] → F (u, v). In [21,Proposition 4.4], the map on posets, P → F (P ), is shown to be a morphism of Hopf algebras. The same holds for the map defined by F .…”

Section: The R-quasisymmetric Function Of a Bruhat Interval And The Cmentioning

confidence: 99%

Quasisymmetric functions and Kazhdan-Lusztig polynomials

Billera

Brenti

2011

Isr. J. Math.

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Abstract. We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.

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On Posets and Hopf Algebras

Cited by 163 publications

References 7 publications

Evolution on distributive lattices

Evolution on distributive lattices

Noncrossing Partitions in Surprising Locations

Quasisymmetric functions and Kazhdan-Lusztig polynomials

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