Log-concavity, q-analogs and the Exponential Formula

1 Introduction 1
2 Basic constructions and notation 8
2.1 Sets and set partitions .................................................................. 9
2.2 Partitions ........................................................................................ 10
2.3 The symmetric group ..................................................................... 15
2.4 Symmetric functions ..................................................................... 18
2.5 Unimodality and log-concavity ......................................................... 24
2.6 Polya frequency sequences ............................................................... 27
3 Rook Theory 31
3.1 Introduction ........................................................................................ 31
3.2 Rook numbers ..................................................................................... 33
3.3 The Foata-Schiitzenberger involution ............................................ 42
3.4 The factorial polynomials ............................................................... 45
3.5 Involution versus factorial polynomials ......................................... 49
3.6 A q-analog of the rook num bers ...................................................... 49
3.7 The involution preserves the statistic ............................................. 51
3.8 Strong log-concavity of g-rook numbers ......................................... 53
4 Log-concavity 58
4.1 Bender and Canfield’s main result ................................................... 58
4.2 Applications ........................................................................................ 60
4.3 Main Theorem .................................................................................. 62
4.4 Applications to polynomial sequences ............................................ 65
4.5 Proof of the Main T heorem ............................................................ 70
4.6 Iteration of the cycle index operator ............................................... 77
4.7 Parametrized log-concavity ............................................................... 82
4.8 Weighted Jacobi-Trudi m atrices ..................................................... 92
5 M ixed concavity and convexity 94
5.1 The log-Fibonacci property ........................................................... 94
5.2 Log-Fibonacci sequences .................................................................. 95
5.3 The strong log-Fibonacci property .................................................. 97
5.4 Strongly log-Fibonacci sequences .................................................. 99