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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.
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