The notion of cumulants was extended to monotone convolution by Hasebe and
Saigo. Contrary to the classical, free and Boolean case,
monotone independence does not satisfy the axiom of exchangeability
and thus mixed cumulants do not vanish.
It does however satisfy the weaker axiom of spreadability
and in joint work with T.Hasebe we found that instead of vanishing, the
mixed cumulants involve the coefficients of the Campbell-Baker-Hausdorff
series.
More recently Novelli and Thibon pointed out an abstract interpretation of
these cumulant identities in terms of Eulerian idempotents in the Hopf
algebra WQSym of word quasi-symmetric functions.
We report on our joint effort to understand this correspondence.