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 f-vectors and h-vectors 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 Stanley-Reisner 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 k-equal partition lattice (the intersection lattice of the k-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the k-equal 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.

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

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 0-1 integer programming are often preferred even though they require much modelling effort. To offer a better trade-off "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 n-ary constraints like atmost1-incommon, 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

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...

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.

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...

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Most of the popular interconnection networks can be represented as Cayley graphs. Star graph is one of the extensively studied undirected Cayley graphs, which is considered to be an attractive alternative to the popular binary n-cube. The n-rotator graph and the cycle prefix digraph are a set of directed Cayley graphs introduced recently. Since the recently introduced directed Cayley graphs have some interesting properties, a comparative study of star and directed Cayley graphs is worthy of study. In this paper we compare the structural and algorithmic aspects of star graphs with that of directed Cayley graphs. In the process we present some new results for star graphs and directed Cayley graphs. We present a formula to calculate the number of nodes at any distance from the identity permutation in star graphs. The minimum bisection width of star and rotator graphs is obtained. Partitioning and fault tolerant parameters for both star and directed Cayley graphs are analyzed. The node disjoint parallel paths and hence the upper bound on the fault diameter of rotator graphs are presented. We compare the minimal path routing in star and rotator graphs using simulation results. Broadcasting and embedding in star and directed Cayley graphs are also compared.

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It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.