自由機率論大致上有兩種研究方法，其一為 Bercovici 與 Voiculescu 所建立的分析方法，又稱為自由調和分析，其二為 Nica 與 Speicher 所發展的組合學方法，也被稱為自由組合學。在研究自由乘法卷積方面，自由組合學給予了極大的貢獻。我們整理了一些自由組合學的重要結果，並利用組合學的方法解釋了在自由調和分析中處理自由乘法卷積的工具–S-transform。最後，我們也利用這些自由組合學的工具重新證明了一些最初是使用自由調和分析方法所證明的結果。

There are generally two approaches to studying free probability theory. The first one, established by Bercovici and Voiculescu, is known as free harmonic analysis, which uses analytical methods. The second approach, developed by Nica and Speicher, is known as free combinatorics, which uses combinatorial methods. Free combinatorics has made significant contributions to the study of free multiplicative convolution. We organize some important results from free combinatorics and provide interpretations of the S-transform, a tool used in free harmonic analysis to handle free multiplicative convolution, using combinatorial methods. Finally, we use these tools to re-prove some results that were originally obtained using free harmonic analysis.

摘要 i
Abstract i
致謝 ii
1 Introduction 1
2 Preliminaries of Free probability 2
2.1 Non-commutative probability space and freeness 2
2.2 Free Combinatorics 3
2.2.1 Non-crossing partitions 3
2.2.2 Non-crossing partitions and permutations 9
2.2.3 Catalan number 12
2.2.4 Formal power series 14
2.3 Free cumulants 16
3 Multiplication of free random variables 24
3.1 Cumulants of free mutiplication 24
3.2 Even element 25
3.3 An application 26
4 The S-transforms with combinatorial arguments 29
4.1 The S-transforms 29
4.2 The S-transforms of even elements 34
References 36

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