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Shellable nonpure complexes and posets. I
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of ..."
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Cited by 132 (9 self)
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The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed fvectors and hvectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their StanleyReisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the kequal partition lattice (the intersection lattice of the kequal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the kequal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
Some Probabilistic Aspects Of Set Partitions
 American Mathematical Monthly
, 1996
"... this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper ..."
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Cited by 22 (2 self)
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this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation
Global Reasoning on Sets
 In Proceedings of Workshop on Modelling and Problem Formulation (FORMUL’01). held alongside CP01
, 2001
"... Finite set constraint systems represent a natural choice to model combinatorial configuration problems involving set disjointness, covering or partitioning relations. However, for efficiency reasons, alternative formulations based on Finite Domain or 01 integer programming are often preferred even ..."
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Cited by 19 (1 self)
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Finite set constraint systems represent a natural choice to model combinatorial configuration problems involving set disjointness, covering or partitioning relations. However, for efficiency reasons, alternative formulations based on Finite Domain or 01 integer programming are often preferred even though they require much modelling effort. To offer a better tradeoff "natural formulation"/efficiency we propose to improve the efficiency of set constraint solvers by introducing global reasoning on a class of finite set constraints. These are nary constraints like atmost1incommon, distinct upon sets of known cardinality. In this paper we show how the representation of sets within powersets specified as set intervals allows us to derive some global pruning based on mathematical and combinatorial analysis formulas. They improve greatly the filtering enforced by bound consistency methods, and allow to detect failure at early stages. Preliminary results are illustrated on the ternary Steiner and a generic distinct problems. 1
The Posterior Probability of Bayes Nets with Strong Dependences
 Soft Computing
, 1999
"... Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong ..."
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Cited by 14 (1 self)
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Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map significant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. Arc probabilities are determined by the amount of mutual information the neighbors contribute to a node, is greater than a critical minimum deviance from independence. A Ø 2 approximation for the probability density function of mutual info...
A partial order on the regions of R n dissected by hyperplanes
 Trans. Amer. Math. Soc
, 1984
"... Abstract. We study a partial order on the regions of R " dissected by hyperplanes. This includes a computation of the Mobius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs. 0. Introdu ..."
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Cited by 13 (0 self)
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Abstract. We study a partial order on the regions of R " dissected by hyperplanes. This includes a computation of the Mobius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs. 0. Introduction. Let % — {77,, H2,...,Hk] be a set of hyperplanes in R". Then the components of R " — UHe.xH form a set 9t of open ncells we will call regions. Traditionally 9t has been studied in terms of enumeration, for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in %. For a thorough discussion of this problem see Zaslavsky's
The Möbius transform on symmetric ordered structures and its application to capacities on finite sets
, 2007
"... ..."
The symmetric Sugeno integral
 Fuzzy Sets and Systems
, 2003
"... We propose an extension of the Sugeno integral for negative numbers, in the spirit of the symmetric extension of Choquet integral, also called ˇ Sipoˇs integral. Our framework is purely ordinal, since the Sugeno integral has its interest when the underlying structure is ordinal. We begin by defining ..."
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Cited by 8 (7 self)
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We propose an extension of the Sugeno integral for negative numbers, in the spirit of the symmetric extension of Choquet integral, also called ˇ Sipoˇs integral. Our framework is purely ordinal, since the Sugeno integral has its interest when the underlying structure is ordinal. We begin by defining negative numbers on a linearly ordered set, and we endow this new structure with a suitable algebra, very close to the ring of real numbers. In a second step, we introduce the Möbius transform on this new structure. Lastly, we define the symmetric Sugeno integral, and show its similarity with the symmetric Choquet integral.
Symmetric and asymmetric fuzzy integrals: the ordinal case
 In: Proc. 6th International Conference on Soft Computing
, 2000
"... functions can be defined in two ways, namley the asymmetric integral (usual Choquet integral) and the symmetric integral (also called the ˇ Sipoˇs integral). No such extension has been defined for the Sugeno integral. In this paper, after recalling the case of Choquet integral, we address the case o ..."
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Cited by 7 (6 self)
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functions can be defined in two ways, namley the asymmetric integral (usual Choquet integral) and the symmetric integral (also called the ˇ Sipoˇs integral). No such extension has been defined for the Sugeno integral. In this paper, after recalling the case of Choquet integral, we address the case of Sugeno integral, which we define in a purely ordinal framework. 1
Free Indexation: Combinatorial Analysis and a Compositional Algorithm
, 1990
"... The principle known as 'free indexation' plays an important role in the determination of the referential properties of noun phrases in the principleand parameters language framework. First, by investigating the combinatorics of free indexation, we show that the problem of enumerating all possible i ..."
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Cited by 6 (0 self)
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The principle known as 'free indexation' plays an important role in the determination of the referential properties of noun phrases in the principleand parameters language framework. First, by investigating the combinatorics of free indexation, we show that the problem of enumerating all possible indexings requires exponential time. Secondly, we exhibit a provably optimal free indexation algorithm.
Fast Algorithms for Generating Integer Partitions
 International Journal of Computer Mathematics
, 1994
"... 1 We present two new algorithms for generating integer partitions in the standard representation. They generate partitions in lexicographic and antilexicographic order, respectively. We prove that both algorithm generate partitions with constant average delay, exclusive of the output. The performanc ..."
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Cited by 6 (0 self)
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1 We present two new algorithms for generating integer partitions in the standard representation. They generate partitions in lexicographic and antilexicographic order, respectively. We prove that both algorithm generate partitions with constant average delay, exclusive of the output. The performance of all known integer partition algorithms is measured and compared, separately for the standard and multiplicity representation. An empirical test shows that both new algorithms are several times faster than any of previously known algorithms for generating unrestricted integer partitions in the standard representation. Moreover, they are faster than any known algorithm for generating integer partition in the multiplicity representation (exclusive of the output). 1 This research is partially supported by NSERC 2 1. Introduction Given an integer n, it is possible to represent it as the sum of one or more positive integers a i , i.e. n=x 1 + x 2 +...+ x m . This representation is called...