Editing Noncrossing partition (section)

==Role in free probability theory==

The lattice of noncrossing partitions plays the same role in defining [[Cumulant#Free cumulants|free cumulants]] in [[free probability]] theory that is played by the lattice of ''all'' partitions in defining joint cumulants in classical [[probability theory]].  To be more precise, let <math>(\mathcal{A},\phi)</math> be a [[non-commutative probability space]] (See [[free probability]] for terminology.), <math>a\in\mathcal{A}</math> a [[non-commutative random variable]] with free cumulants <math>(k_n)_{n\in\mathbb{N}}</math>. Then

:<math>\phi(a^n) = \sum_{\pi\in\text{NC}(n)} \prod_{j} k_j^{N_j(\pi)}</math>

where <math>N_j(\pi)</math> denotes the number of blocks of length <math> j</math> in the non-crossing partition <math>\pi</math>.
That is, the moments of a non-commutative random variable can be expressed as a sum of free cumulants over the sum non-crossing partitions. This is the free analogue of the [[Cumulant#Cumulants and set-partitions|moment-cumulant formula]] in classical probability.
See also [[Wigner semicircle distribution]].