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207
A Logic for Reasoning about Probabilities
 Information and Computation
, 1990
"... We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable ( ..."
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Cited by 213 (20 self)
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We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by DempsterShafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satistiability is NPcomplete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.
Shellable and CohenMacaulay partially ordered sets
 Trans. Amer. Math. Soc
, 1980
"... Abstract. In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finit ..."
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Cited by 118 (5 self)
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Abstract. In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley's main theorem on the JordanHolder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of CohenMacaulay posets. For instance, the lattice of subgroups of a finite group G is CohenMacaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable. Introduction. A pure finite simplicial complex A is said to be shellable if its maximal faces can be ordered F,, F2,..., Fn in such a way that Fk n ( U *j / Fj) is
Combinatorial Hopf algebras and generalized DehnSommerville relations
, 2003
"... A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the u ..."
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Cited by 65 (16 self)
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasisymmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H, ζ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized DehnSommerville relations. We show that, for H = QSym, the generalized DehnSommerville relations are the BayerBillera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the MalvenutoReutenauer Hopf algebra of permutations, the
On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, nonRadon partitions, and orientations of graphs
 TRANS. AMER. MATH. SOC
, 1983
"... ..."
Structure of the MalvenutoReutenauer Hopf algebra of permutations
 Adv. Math
"... Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product ..."
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Cited by 51 (15 self)
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Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasisymmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasisymmetric functions in terms of the facial structure of the cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed by permutations. Our results are obtained from a combinatorial description of the Hopf algebraic structure with respect to a new basis for this algebra, related to the original one via Möbius inversion on the weak order on the symmetric groups. This is in analogy with the relationship between the monomial and fundamental bases of the algebra of quasisymmetric functions. Our results reveal a close relationship between the structure of the MalvenutoReutenauer Hopf algebra and the weak order on the symmetric groups.
Free Probability Theory And NonCrossing Partitions
 LOTHAR. COMB
, 1997
"... Voiculescu's free probability theory  which was introduced in an operator algebraic context, but has since then developed into an exciting theory with a lot of links to other fields  has an interesting combinatorial facet: it can be described by the combinatorial concept of multiplicative f ..."
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Cited by 45 (4 self)
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Voiculescu's free probability theory  which was introduced in an operator algebraic context, but has since then developed into an exciting theory with a lot of links to other fields  has an interesting combinatorial facet: it can be described by the combinatorial concept of multiplicative functions on the lattice of noncrossing partitions. In this survey I want to explain this connection  without assuming any knowledge neither on free probability theory nor on noncrossing partitions.
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
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Cited by 45 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
A Class of Geometric Lattices Based on Finite Groups
, 1972
"... For any finite group G and positive integer n a finite geometric 1attice Q (G) of rank n, the lattice of partial Gpartitions, is constructed. n ..."
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Cited by 45 (0 self)
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For any finite group G and positive integer n a finite geometric 1attice Q (G) of rank n, the lattice of partial Gpartitions, is constructed. n
An axiomatic approach to the concept of interaction among players in cooperative games
 Int. Journal of Game Theory
, 1999
"... version remanie: le 8/10/98 An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, ..."
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Cited by 42 (26 self)
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version remanie: le 8/10/98 An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, we give an expression of the Banzhaf and Shapley interaction indices in terms of pseudoBoolean functions. 1
On Sugeno integrals as an aggregation function
 Fuzzy Sets and Systems
, 2000
"... The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggregation. In particular, we give some equivalent expressions of it. We also give an axiomatic characterization of the class of all the Sugeno integrals. Some particular subclasses, such as the weighted maximum and m ..."
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Cited by 35 (9 self)
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The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggregation. In particular, we give some equivalent expressions of it. We also give an axiomatic characterization of the class of all the Sugeno integrals. Some particular subclasses, such as the weighted maximum and minimum functions are investigated as well.