Cumulants, free cumulants and half-shuffles
Résumé
Free cumulants were introduced as the proper analog of class
ical cumulants in the
theory of free probability. There is a mix of similarities an
d differences, when one considers the
two families of cumulants. Whereas the combinatorics of cla
ssical cumulants is well expressed in
terms of set partitions, the one of free cumulants is describ
ed, and often introduced in terms of
non-crossing set partitions. The formal series approach to
classical and free cumulants also largely
differ.
It is the purpose of the present article to put forward a differ
ent approach to these phenomena.
Namely, we show that cumulants, whether classical or free, c
an be understood in terms of the algebra
and combinatorics underlying commutative as well as non-co
mmutative (half-)shuffles and (half-
)unshuffles. As a corollary, cumulants and free cumulants can
be characterized through linear fixed
point equations. We study the exponential solutions of thes
e linear fixed point equations, which
display well the commutative, respectively non-commutati
ve, character of classical, respectively
free, cumulants.