Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined for the surfaces S_0, S_1, N_1,and N_2. An algorithm is presented for generating the irreducible triangulations of a fixed surface using triangulations of other surfaces. This algorithm has been implemented as a computer program which terminates for S_1, S_2, N_1, N_2, N_3, and N_4. Thus the complete sets irreducible triangulations are now also known for S_2, N_3, and N_4, with respective cardinalities 396784, 9708, and 6297982.
A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$ vertices. The best previous bound was $171g-72$.
The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of these irreducible triangulations.
We give the complete list of the 29 irreducible triangulations of the Klein bottle. We show how the construction of Lawrencenko and Negami, which listed only 25 such irreducible triangulations, can be modified at two points to produce the 4 additional irreducible triangulations of the Klein bottle.
A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with precisely one boundary component. By implementing the algorithm on computer, we have found a list of 297 nonisomorphic combinatorial types of irreducible triangulations on the once-punctured torus.
This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For…
A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boun…
This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific "transversal structures" on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are re…
A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations.
We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2) triangulations with all vertices of degree at least 4. The method is based on a series of reversible operations, termed reductions, which lead to a minimal set of triangulations in each family. Throughout the process the triangulations remain within the corresponding family. Moreover, for the family (1) these operations reduce to the well-known edge contraction…
A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible triangulations of any punctured surface is established. Complete lists of irreducible triangulations are determined for the Möbius band (6 in number) and the pinched torus (2 in number). All the non-isomorphic combinatorial types (20 in number) of triangulations of the projective plane with up to 8 vertices are determined.
Let $Φ$ be an irreducible crystallographic root system and $\mathcal P$ its root polytope, i.e., its convex hull. We provide a uniform construction, for all root types, of a triangulation of the facets of $\mathcal P$. We also prove that, on each orbit of facets under the action of the Weyl gruop, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.
We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{λ\in \mathbb{R}^n: \, 0\leq \frac{λ_1}{1}\leq \frac{λ_2}{2}\leq \frac{λ_3}{3}\leq \cdots \leq \frac{λ_n}{n}\right\} . \] We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of…
We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of non-separable triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. Our proofs first follow Tutte's classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration. Then we …
We develop a method to compute the generating function of the number of vertices inside certain regions of the Uniform Infinite Planar Triangulation (UIPT). The computations are mostly combinatorial in flavor and the main tool is the decomposition of the UIPT into layers, called the skeleton decomposition, introduced by Krikun. In particular, we get explicit formulas for the generating functions of the number of vertices inside hulls (or completed metric balls) centered around the root, and the number of vertices inside geodesic slices of these hulls. We also recover known results about the sc…
We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and "local contributions" of the links of its nonunimodular faces. We also compute new examples of nonunimodal h^*-vectors of reflexive polytopes.
We present a new derivation of the distance-dependent two-point function of random planar triangulations. As it is well-known, this function is intimately related to the generating functions of so-called slices, which are pieces of triangulation having boundaries made of shortest paths of prescribed length. We show that the slice generating functions are fully determined by a direct recursive relation on their boundary length. Remarkably, the kernel of this recursion is some quantity introduced and computed by Tutte a long time ago in the context of a global enumeration of planar triangulation…
Let $f_n$ be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides. Suppose ${\mathcal T}_n$ is a random triangulation, sampled uniformly out of all possible triangulations of $P_n$. We study the sum of weights of triangles in ${\mathcal T}_n$ and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of $f_n$ in which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new pr…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials.
We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth…
A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As the objects of Fsk are meaningfully bracketed words in the skew unit I and the generating object X, it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them. This leads us to relationships between triangulations of polygons, the Tamari lattice, left and right bracketing functions, and the orientals. A consequence of ou…
Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P^2-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten tetrahedra.
The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $σ$ has weight $λ^{|σ|}$, where $λ$ is a positive real parameter, and $|σ|$ is the total length of the edges in $σ$. Empirically, this model exhibits a "phase transition" at $λ=1$ (corresponding to the uniform distribution): f…
We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of NC(n). As a consequence, we show that the even generating function for irreducible meandric systems is the R-transform of XY, where X and Y are classically (commuting) independent random variables, and each of X,Y has centred semicircular distribution of variance 1. Following this point of view, we make some observations about the symmetric linear functional on polynomials which has R-transform given by the even generating function for meanders.
In this paper we study canonical $γ$-structures, a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A $γ$-structure is composed by specific building blocks, that have topological genus less than or equal to $γ$, where composition means concatenation and nesting of such blocks. Our main result is the derivation of the generating function of $γ$-structures via symbolic enumeration using so called irreducible shadows. We furthermore recursively compute the generating polynomials of irreducible shadows of genus $\le γ$…
The hull perimeter at distance d in a planar map with two marked vertices at distance k from each other is the length of the closed curve separating these two vertices and lying at distance d from the first one (d<k). We study the statistics of hull perimeters in large random planar triangulations and quadrangulations as a function of both k and d. Explicit expressions for the probability density of the hull perimeter at distance d, as well as for the joint probability density of hull perimeters at distances d1 and d2, are obtained in the limit of infinitely large k. We also consider the situa…
Although degree bounds and algorithms for the generators of various invariant rings have been known for decades, little is known about the cardinality of minimal generating sets. Estimates of such would provide lower bounds for the runtime of algorithms that compute invariants. Fix a semisimple linear algebraic group, choose an irreducible representation of highest weight w, and consider the irreducible representations of highest weight nw. As n goes to infinity, the cardinality of a minimal set of generating invariants grows faster than any polynomial in n. On the other hand, combinatorial me…
