Cumulants, free cumulants and half-shuffles

Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, that of free cumulants is described and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differs. The purpose of this study is to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (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 these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical and free cumulants.

A New Look at the Coefficients of a Reciprocal Generating Function

We study a special property of free cumulants. We prove that coefficients of a reciprocal generating function correspond to “free cumulants with the first two elements in the same block.”

Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps

The q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of itsmoments in terms ofmatchings, where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case q = 0 of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

COMBINATORIAL RELATION BETWEEN FREE CUMULANTS AND JACOBI PARAMETERS

We find a formula which expresses free and conditionally free cumulants in terms of Jacobi parameters. This leads to some necessary conditions for free and conditionally free infinite divisibility. We also express conditionally free cumulants of two measures in terms of their free cumulants.