Locally Oriented Noncrossing Trees

Abstract

We define an orientation on the edges of a noncrossing tree induced by the labels: for a noncrossing tree (i.e., the edges do not cross) with vertices $1,2,\ldots,n$ arranged on a circle in this order, all edges are oriented towards the vertex whose label is higher. The main purpose of this paper is to study the distribution of noncrossing trees with respect to the indegree and outdegree sequence determined by this orientation. In particular, an explicit formula for the number of noncrossing trees with given indegree and outdegree sequence is proved and several corollaries are deduced from it. Sources (vertices of indegree $0$) and sinks (vertices of outdegree $0$) play a special role in this context. In particular, it turns out that noncrossing trees with a given number of sources and sinks correspond bijectively to ternary trees with a given number of middle- and right-edges, and an explicit bijection is provided for this fact. Finally, the in- and outdegree distribution of a single vertex is considered and explicit counting formulas are provided again.

Refined enumeration of 2-noncrossing trees

“…for the number of noncrossing trees on n vertices with i sources. This formula was first obtained by Okoth and Wagner in [5].…”

“…Vertex 3 is a white source while vertex 2 is a non-source white vertex. The number of noncrossing trees, having a local orientation, with a given number of sources and sinks was obtained by the present author and his co-author in [5]. These trees are called locally oriented noncrossing trees or lnc-trees therein.…”

Bijections of plane Husimi graphs and certain combinatorial structures

“…In Figure 1, we show a plane Husimi graph on 11 vertices with 6 blocks. Plane Husimi graphs as well as other related structures have been extensively studied in [1,2,4,5,7,[9][10][11][12][13]16] among other papers. A block b attached to a vertex i in a plane Husimi graph is called a block child of i and the outdegree of a vertex j is the number of its block children [13].…”