Unimodal Log-Concave and Polya Frequency Sequences in Combinatorics.

Intro -- Contents -- Foreword -- Introduction -- Chapter 1 The Poset Conjecture -- 1.1 Introduction -- 1.2 Statement of the Poset Conjecture -- 1.3 Partial results on the Poset Conjecture -- 1.4 A second look at the Poset Conjecture -- Chapter 2 A General Theory -- 2.1 Introduction -- 2.2 Polynomials with only real zeros and total positivity -- 2.3 The six fundamental bases and the transition matrices between them -- 2.4 Linear transformations that preserve the PF property -- 2.5 Linear transformations that preserve the PF[sub(2)] property -- 2.6 Summary of results and open problems -- Chapter 3 Ramifications of the General Theory -- 3.1 Introduction -- 3.2 Further properties of the six fundamental bases -- 3.3 The special case of polynomials with no constant term -- 3.4 Eulerian, Lagrange, and Krawtchouk polynomials -- 3.5 Three general problems -- Chapter 4 Polynomials in PF[([sup(x+d-i)][sub(d)])] -- 4.1 Introduction -- 4.2 Elementary properties of PF[([sup(x+d-i)][sub(d)])] -- 4.3 A Fundamental Theorem -- 4.4 Consequences of the Fundamental Theorem -- 4.5 The characterization of PF-sequences -- 4.6 The characterization of PF[([sup(x+d-i)][sub(d)])] -- 4.7 Products of polynomials in PF[([sup(x+d-i)][sub(d)])] -- Chapter 5 Applications to the Poset Conjecture -- 5.1 Introduction -- 5.2 Ferrers posets -- 5.3 Column strict labeled Ferrers posets and the solution of a conjecture -- 5.4 A class of naturally labeled Ferrers posets -- 5.5 Disjoint unions of chains -- 5.6 Gaussian posets -- 5.7 Necessary and sufficient conditions for the Poset Conjecture -- Chapter 6 Applications to Enumerative Combinatorics -- 6.1 Introduction -- 6.2 PF sequences arising from symmetric functions and Jack polynomials -- 6.3 Zeta polynomials of partially ordered sets -- 6.4 Functions of a finite set into itself -- 6.5 Associated Lah numbers -- 6.6 Stirling permutations.

6.7 Associated Stirling numbers -- 6.8 Colorings of graphs -- Chapter 7 Polya Frequency Digraphs -- 7.1 Introduction -- 7.2 PF digraphs and the distributive lattice conjecture -- 7.3 A general result -- 7.4 The connection with the theory of symmetric functions -- 7.5 The inversion Theorem -- 7.6 PF[sub(2)] digraphs -- 7.7 Semitransitive digraphs -- Bibliography -- Appendix -- Tables.

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