Andre Permutations, Lexicographic Shellability and the cd-Index of a Convex Polytope

Purtill, Mark

doi:10.2307/2154445

Cited by 10 publications

(15 citation statements)

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“…P Since C n+1 = Prism(C n ), Proposition 4.2 gives a recursion formula for the cd-index of the cubical lattice C n . This recursion was first developed by Purtill [17]. The second part of Proposition 4.2 may be generalized in the following manner.…”

Section: Proposition 42 Let P Be a Graded Poset Thenmentioning

confidence: 99%

“…For instance, the cd-index of B n is a refined enumeration of André permutations [17]. Similarly, it is also a refined enumeration of simsun permutations, first defined by Simion and Sundaram [22,23].…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, it is also a refined enumeration of simsun permutations, first defined by Simion and Sundaram [22,23]. Another known example of a poset-permutations pair is the cubical lattice and signed André permutations [8,17]. This motivates us to ask the following question.…”

Section: Introductionmentioning

confidence: 99%

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Ehrenborg

Readdy

1998

Journal of Algebraic Combinatorics

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Abstract. The linear span of isomorphism classes of posets, P, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space P to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the ab-index of the Cartesian product of two posets given the ab-indexes of each poset.

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Section: Proposition 42 Let P Be a Graded Poset Thenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ehrenborg

Readdy

1998

Journal of Algebraic Combinatorics

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“…It embodies in an elegant way the linear relations of flag vectors of Eulerian posets (the generalized Dehn-Sommerville relations of Bayer and Billera [1]); the number of coefficients in the cd-index is a Fibonacci number. It is known to be nonnegative for polytopes (see Stanley [12]), but it is not known what it counts, except in special cases (see Purtill [11]). Among polytopes, the cd-index is minimized by the simplices (see Billera and Ehrenborg [5]).…”

Section: Introductionmentioning

confidence: 99%

Signs in the 𝑐𝑑-index of Eulerian partially ordered sets

Bayer

2000

Proc. Amer. Math. Soc.

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Abstract. A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative cd-index. This paper completely determines which entries of the cd-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on cd-coefficients (except for the coefficient of c n ).

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“…The first recursion formulas for the cd-index were given by Purtill [21] for the Boolean algebra and the cubical lattice, that is, the face lattice of the n-simplex and the n-cube. In [16] the authors gave shorter recursions using derivations, as well as determined how the cd-index changes under the pyramid and prism operations.…”

Section: Introductionmentioning

confidence: 99%

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

Ehrenborg¹,

Readdy²

1999

Discrete Comput Geom

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Abstract. We study complex hyperplane arrangements whose intersection lattices, known as the Dowling lattices, are a natural generalization of the partition lattice. We give a combinatorial description of the Dowling lattice via enriched partitions to obtain an explicit EL-labeling and then find a recursion for the flag h-vector in terms of weighted derivations. When the hyperplane arrangements are real they correspond to the braid arrangements A n and B n . By applying a result due to Billera and the authors, we obtain a recursive formula for the cd-index of the lattice of regions of the braid arrangements A n and B n .

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Andre Permutations, Lexicographic Shellability and the cd-Index of a Convex Polytope

Cited by 10 publications

References 0 publications

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Signs in the 𝑐𝑑-index of Eulerian partially ordered sets

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

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