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 proposi-tional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatiza-tion and show that the problem of deciding satistiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatiza-tions, we give a complete axiomatization for reasoning about Boolean combina-tions 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.

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 quasi-symmetric functions; this explains the ubiquity of quasi-symmetric 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 Dehn-Sommerville relations. We show that, for H = QSym, the generalized Dehn-Sommerville relations are the Bayer-Billera 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 Malvenuto-Reutenauer Hopf algebra of permutations, the

For any finite group G and positive integer n a finite geometric 1attice Q (G) of rank n, the lattice of partial G-partitions, is constructed. n

Abstract. We analyze the structure of the Malvenuto-Reutenauer 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 quasi-symmetric 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 quasi-symmetric 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 quasi-symmetric functions. Our results reveal a close relationship between the structure of the Malvenuto-Reutenauer Hopf algebra and the weak order on the symmetric groups.

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 non-crossing partitions. In this survey I want to explain this connection -- without assuming any knowledge neither on free probability theory nor on non-crossing partitions.

Abstract. We consider a specialization YM(q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M. We show that the coefficients of YM(1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of M. In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid. 0. Introduction. In 1954, Tutte [30] introduced the dichromate of a (finite) graph, which has since become known as the Tutte polynomial. In the four decades since then this has provided a profound link between combinatorics and other branches of mathematics

As proposed in various places, a set of propositional formulas, each associated with a numerical weight, can be used to model the preferences of an agent in combinatorial domains. If the range of possible choices can be represented by the set of possible assignments of propositional symbols to truth values, then the utility of an assignment is given by the sum of the weights of the formulas it satisfies. Our aim in this paper is twofold: (1) to establish correspondences between certain types of weighted formulas and well-known classes of utility functions (such as monotonic, concave or k-additive functions); and (2) to obtain results on the comparative succinctness of different types of weighted formulas for representing the same class of utility functions.

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the Mobius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a generalization of Stanley's theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. We also give some applications of this second main theorem, including the Tamari lattices.

this paper we describe the structure of the group M 1 (Theorem 1.6, Corollary 1.7). Quite surprisingly, it turns out to be possible to do this via a "transform" which converts the convolution of multiplicative functions into the multiplication of formal power series (in the same way as the convolution of functions in L

A classical-like combinatorial interpretation of the Fibonomial coefficients is provided following [1,2]. An adequate combinatorial interpretation of recurrence satisfied by Fibonomial coefficients is also proposed. It is considered to be- in the spirit classical- combinatorial interpretation like binomial Newton and Gauss q-binomial coefficients or Stirling number of both kinds are. (See ref. [3,4] and refs. given therein). It also concerns choices. Choices of specific subsets of maximal chains from a non-tree poset specifically obtained starting from the Fibonacci rabbits ‘ tree. Several figures illustrate the exposition of statements- the derivation of the recurrence itself included. 1