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Multiplicative functions on the lattice of non-crossing partitions and free convolution. (English) Zbl 0791.06010
We examine the lattice of non-crossing partitions both from a combinatorial and from a probabilistic point of view. In the first part, we consider multiplicative functions on this lattice and describe the convolution of such functions by generating power series. This is used for deriving some known results on non-crossing partitions in a unified and short way. In the second part, we work out the connection between the lattice of non-crossing partitions and the ‘free convolution’ of probability measures in the sense of Voiculescu. A new combinatorial proof of Voiculescu’s main formula for this convolution is given.

MSC:
06A07 Combinatorics of partially ordered sets
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
60C05 Combinatorial probability
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