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By Roland Speicher

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Aim−1 +1 , . . , aim )}. 3. 2 where τ = 1m . Then the statement of the theorem is the following: Consider m ∈ N, an increasing sequence 1 ≤ i1 < i2 < · · · < im := n and random variables a1 , . . , an . Put σ := {(a1 , . . , ai1 ), . . , (aim−1 +1 , . . , aim )}. Then we have: km [a1 · · · ai1 , . . , aim−1 +1 · · · aim ] = (101) kπ [a1 , . . , an ] . e. kn−1 (a1 , . . , am−1 , am · am+1 , am+2 , . . , an ). In that case σ = {(1), (2), . . , (m, m+1), . . , (n)} and it is easily seen that the partitions π ∈ N C(n) with the property π ∨ σ = 1n are either 1n or those non-crossing partitions which consist of two blocks such that one of them contains m and the other one contains m + 1.

6. 1) Note that the operation (f, g) → f ∗ g is commutative and associative. 2) According to the theory of M¨obius inversion F is a bijective mapping whose inverse is given by (106) F −1 (f )(π) = f (σ)µ(σ, π). σ∈P σ≤π 3) Denote by 1σ and 1≥σ the functions given by (107) 1σ (π) = 1, if π = σ 0, otherwise 1≥σ (π) = 1, if π ≥ σ 0, otherwise. and (108) Then we have F (1σ )(π) = 1σ (τ ) = 1≥σ (π), τ ≤π and hence F (1σ ) = 1≥σ . 7. Let P be a finite lattice. Then we have for arbitrary functions f, g : P → C that F (f ∗ g) = F (f ) · F (g), (109) where on the right hand side we have the pointwise product of functions.

5. Consider the following examples for the bijection between Catalan paths and non-crossing pair partitions. • n=2 ✒❅ ❘ ❅ = ˆ • n=4 ✒❅ ✒❅ ❘ ❅ ❘ ❅ = ˆ ✒❅ ❘ ❅ ❅ ❘ ❅ = ˆ ✒ 42 3. 6. 1) According to the free central limit theorem the semi-circular distribution has to be considered as the free analogue of the normal distribution and is thus one of the basic distributions in free probability theory. 2) As in the classical case the assumptions in the free central limit theorem can be weakened considerably.

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