By Roland Speicher

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**Extra resources for Combinatorics of free probability theory [Lecture notes]**

**Sample text**

N ¯) = ˆ N C(n) with π ∪ σ ∈ N C(1, ¯1, . . , n, n ¯) . Example: Consider the partition π := {(1, 2, 7), (3), (4, 6), (5), (8)} ∈ N C(8). For the complement K(π) we get K(π) = {(1), (2, 3, 6), (4, 5), (7, 8)} , as can be seen from the graphical representation: 1 ¯1 2 ¯2 3 ¯3 4 ¯4 5 ¯5 6 ¯6 7 ¯7 8 ¯8 . ], ... 5. 1) Denote by Sn the group of all permutations of (1, 2, . . , n). Let α be a permutation in Sn , and let π = {V1 , . . , Vr } be a partition of (1, 2, . . , n). Then α(V1 ), . . , α(Vr ) form a new partition of (1, 2, .

N}) is a partition of {1, . . , n}\I. If there does not exist such a decomposition of π, then we call π indecomposable. A function t : n∈N P(N ) → C is called multiplicative, if we have for each decomposition π = π1 ∪ π2 as above that t(π1 ∪ π2 ) = t(π1 ) · t(π2 ) (where we identify of course P({k, k + 1, . . , k + r}) with P(r + 1)). Consider now a random variable a whose moments are given by the formula (98) ϕ(an ) = t(π), π∈P(n) where t is a multiplicative function on the set of all partitions.

Vr | = 2|Wr |. 3. 1. Motivated by our combinatorial description of the free central limit theorem we will in the following use the non-crossing partitions to write moments ϕ(a1 · · · an ) in the form π∈N C(n) kπ [a1 , . . , an ], where k denotes the so-called free cumulants. Of course, we should invert this equation in order to define the free cumulants in terms of the moments. This is a special case of the general theory of M¨obius inversion and M¨obius function – a unifying concept in modern combinatorics which provides a common frame for quite a lot of situations.

### Combinatorics of free probability theory [Lecture notes] by Roland Speicher

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