Noncommutative crossing partitions

We define and study noncommutative crossing partitions which are a generalization of non-crossing partitions. By introducing a new cover relation on binary trees, we show that the partially ordered set of noncommutative crossing partitions is a graded lattice. This new lattice contains the Kreweras lattice, the lattice of non-crossing partitions, as a sublattice. We calculate the M\"obius function, the number of maximal chains and the number of $k$-chains in this new lattice by constructing an explicit $EL$-labeling on the lattice. By use of the $EL$-labeling, we recover the classical results on the Kreweras lattice. We characterize two endomorphism on the Kreweras lattice, the Kreweras complement map and the involution defined by Simion and Ullman, in terms of the maps on the noncommutative crossing partitions. We also establish relations among three combinatorial objects: labeled $k+1$-ary trees, $k$-chains in the lattice, and $k$-Dyck tilings.Comment: 45 page

Random matrices, continuous circular systems and the triangular operator

We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of a formula for *-moments of the triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.Comment: 26 pages, 5 figures, 1 reference added, minor change

Relations between cumulants in noncommutative probability

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.Comment: 27 pages, 7 figures, AMS LaTe