Polynomial maps with applications to combinatorics and probability theory

2 Acknowledgments 3 Introduction 7 Chapter I. Exponential Bell polynomials 11 1. Properties 12 2. Moments and cumulants 14 3. Sums of independent random variables 18 4. Circular processes and partitions of an n-set 20 5. Further Properties of En 36 6. Measures on Poisson lattices 41 Chapter II. Ordinary Bell polynomials 45 1. Properties 45 2. Recurrent events and the renewal equation 47 3. Shift polynomials, Hankel mean-independence 48 4. Moment sequence preserving maps, ordinary cumulants 50 Chapter III. Comtet polynomials and binomial posets 55 1. Properties 55 2. Recurrent events and binomial posets 56 3. Characterization of Hankel mean-independence 59 4. Moment sequence preserving maps 62 Chapter IV. Compound polynomials 65

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