A sequence of real numbers {a}fn=o,1,2,... is said to be a P6lya frequency sequence of order r (or, a PF, sequence, for short) if, for all 1 < t < r, the determinants of all the minors of order t of the infinite matrix (ayi)ij=o,1,2,... (where ak ' 0 if k < 0) are nonnegative. Clearly PF 2 =a log-concave and PF = PF 2 =. unimodal. It is known that a sequence {ao, a,,...,ad, 0,0,...} is PF if and only if the polynomial EZ a xi has nonnegative coefficients and only real zeros.Many sequences in Combinatorics are known to be unimodal, PF 2 or PF. The object of this thesis is to develop general methods which generate new unimodal, PF 2 or PF sequences from existing ones, and to apply such methods to prove the unimodality, PF 2 or PF property of sequences of combinatorial interest.In the first part of this thesis we consider the vector space Vd of all polynomials having real coefficients and degree < d.

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We introduce and study three new statistics on the hyperoctahedral group B n and show that they give two generalizations of Carlitz's identity for the descent number and major index over S n . This answers a question posed by Foata.  2001 Elsevier Science

Abstract. For a simplicial complex or more generally Boolean cell complex ∆ we study the behavior of the f -and h-vector under barycentric subdivision. We show that if ∆ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney-Davis conjecture for spheres that are the subdivision of a Boolean cell complex.For a general (d − 1)-dimensional simplicial complex ∆ the hpolynomial of its n-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this hpolynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d.

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Abstract. In this paper we present several results and open problems about logconcavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial.

We consider the location of zeros of four related classes of polynomials, one of which is the class of chromatic polynomials of graphs. All of these polynomials are generating functions of combinatorial interest. Extensive calculations indicate that these polynomials often have only real zeros, and we give a variety of theoretical results which begin to explain this phenomenon. In the course of the investigation we prove a number of interesting combinatorial identities and also give some new sufficient conditions for a polynomial to have only real zeros.