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FRANCESCO BRENTI

examined thoroughly in this work. However, as we will show, these already provide suf-

ficiently powerful tools to solve a vast number of combinatorial problems that had so far

resisted solution by other techniques.

The organization of the paper is as follows. In Chapter 1 we present, as a combinatorial

motivation, a remarkable and still open unimodality conjecture due to R. Stanley. It was

the study of this problem that originally led the author to the consideration of linear trans-

formations that preserve the P F property of a sequence. In Chapters 2, 3 and 4 we study

in detail several linear transformations that are often used in mathematics, and especially

in combinatorics, and their effect on Polya frequency sequences. These chapters form the

theoretical core of the paper and can be read independently from the other chapters by the

reader uninterested in their combinatorial applications. They also contain many open prob-

lems and conjectures. In Chapters 5, 6 and 7 the theoretical "machinery" developed in the

preceding three chapters is applied to several combinatorial situations. More precisely, in

Chapter 5 we verify the conjecture presented in Chapter 1 in several remarkable cases, thus

also providing, in particular, a proof of another conjecture by R. Stanley. Chapter 6 is, in-

stead, devoted to other applications to enumerative combinatorics, these include applications

to plane partitions, zeta polynomials of partially ordered sets, colorings of graphs, functions

of a finite set into itself, associated Lah numbers, Stirling permutations and polynomials,

Ward and Jordan numbers. Finally, in Chapter 7, we show that the problem presented in

Chapter 1 is actually only a special case of a more general problem that can be studied with

the techniques developed in Chapters 2, 3 and 4 and that is also connected to the theory of

symmetric functions.