Enumeration of noncrossing trees on a circle

AbstractWe consider several enumerative problems concerning labelled trees whose vertices lie on a circle and whose edges are rectilinear and do not cross

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric functions arising from parking functions. We close by presenting an alternative proof of our product formulas communicated to us by Christian Krattenthaler which uses generating functions and Lagrange inversion

Dyck paths with coloured ascents

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon, etc. In some cases enumeration gives new expression for sequences enumerating these structures.Comment: 14 pages, 11 figure

A self-dual poset on objects counted by the Catalan numbers and a type-B analogue

We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of $P^A_n$ and $P^B_n$ are subsets of the symmetric and the hyperoctahedral groups, consisting of permutations which avoid certain patterns. The order relation is given by (strict) containment of the descent sets. In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions. Several structural properties of these permutation posets follow, including self-duality and the strong Sperner property. We also discuss posets $Q^A_n$ and $Q^B_n$ similarly associated with noncrossing partitions, defined by means of the excedence sets of suitable pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure

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