Non-crossing tree realizations of ordered degree sequences

We investigate the enumeration of non-crossing tree realizations of integer sequences, and we consider a special case in four parameters, that can be seen as a four-dimensional tetrahedron that generalizes Pascal's triangle and the Catalan numbers

Graph realizations constrained by skeleton graphs

In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM) model to capture the fundamental difference in assortativity of networks in nature studied by the physical and life sciences and social networks studied in the social sciences. In 2014 Czabarka proposed a direct generalization of the JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the vertices have specified degrees, and the vertex set itself is partitioned into classes. For each pair of vertex classes the number of edges between the classes in a graph realization is prescribed. In this paper we apply the new {\em skeleton graph} model to describe the same information as the PAM model. Our model is more convenient for handling problems with low number of partition classes or with special topological restrictions among the classes. We investigate two particular cases in detail: (i) when there are only two vertex classes and (ii) when the skeleton graph contains at most one cycle.Comment: 19 page

Order-Invariant First-Order Logic over Hollow Trees

We show that the expressive power of order-invariant first-order logic collapses to first-order logic over hollow trees. A hollow tree is an unranked ordered tree where every non leaf node has at most four adjacent nodes: two siblings (left and right) and its first and last children. In particular there is no predicate for the linear order among siblings nor for the descendant relation. Moreover only the first and last nodes of a siblinghood are linked to their parent node, and the parent-child relation cannot be completely reconstructed in first-order

Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of presentatio

Associahedra via spines

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the classical permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction

Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.Comment: 51 page

On the Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.88^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety $CM^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n-4$ hyperplanes yields at most $deg(CM^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}={1/2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular we show that it leads to an upper bound of $2 D^{3,n}= {\frac{2^{n-3}}{n-2}}{{n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions