In 1978, Uffe Haagerup proved an important inequality relating two
functional analytic norms (the $\ell^2$-norm and the convolution norm) on
the group algebra of a free group. While invented to be used in operator
algebras, the inequality also found important uses in geometric group
theory, probability theory, and other fields.

From a combinatorial standpoint, one can analyze the above-mentioned norms
in terms of {\em free cumulants} associated to elements in the free group.
Free cumulants are multilinear functions defined using M\"obius inversion
on the lattice of non-crossing partitions $NC$.

In this talk, I will discuss recent work on a surprisingly strong
improvement of Haagerup's inequality, which extends far beyond the context
of the free group. The results rely fundamentally on the structure of
$NC$.

A working knowledge of partitions and basic combinatorial constructions
such as M\"obius inversion will be an asset, but {\em no knowledge of
functional analysis} will be assumed.