Annals of Combinatorics

Adams, Colin; Shinjo, Reiko; Tanaka, Kokoro

doi: 10.1007/s00026-011-0109-2pmid: N/A

An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links: (3, 5, 7, . . .), (2, n, n + 1, n + 2, . . .) for each n ≥ 3, (3, n, n + 1, n + 2, . . .) for each n ≥ 4. Moreover, the finite sequences (2, 4, 5) and (3, 4, n) for each n ≥ 5 are universal for all knots and links. It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.

Avelino, Catarina; Santos, Altino

doi: 10.1007/s00026-011-0110-9pmid: N/A

Dihedral f-tilings by spherical parallelograms and spherical triangles were obtained in [3–5]. In this paper we extend these results presenting the study of all dihedral f-tilings of the sphere S 2, whose prototiles are a spherical equilateral or isosceles triangle and a spherical isosceles trapezoid. The combinatorial structure, including the symmetry group of each tiling, is given in Table 1.

Bonato, Anthony; Janssen, Jeannette

doi: 10.1007/s00026-011-0111-8pmid: N/A

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability $${p \in (0, 1)}$$ if the metric distance between the vertices is below a given threshold. For certain choices of V as a countable dense set in $${\mathbb{R}^n}$$ equipped with the metric derived from the L ∞-norm, it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR n , is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR n . In contrast, we show that infinite random geometric graphs in $${\mathbb{R}^{2}}$$ with the Euclidean metric are not necessarily isomorphic.

Bonin, Joseph

doi: 10.1007/s00026-011-0112-7pmid: N/A

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzík proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r ≥ 3, no rank−r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.

Chow, Chak-On; Shiu, Wai

doi: 10.1007/s00026-011-0113-6pmid: N/A

We count in the present work simsun permutations of length n by their number of descents. Properties studied include the recurrence relation and real-rootedness of the generating function of the number of n-simsun permutations with k descents. By means of generating function arguments, we show that the descent number is equidistributed over n-simsun permutations and n-André permutations. We also compute the mean and variance of the random variable X n taking values the descent number of random n-simsun permutations, and deduce that the distribution of descents over random simsun permutations of length n satisfies a central and a local limit theorem as n → + ∞.

Conflitti, Alessandro; Mamede, Ricardo

doi: 10.1007/s00026-011-0114-5pmid: N/A

We present an explicit bijection between noncrossing and nonnesting partitions of Coxeter systems of type D which preserves openers, closers, and transients.

Duchamp, Gérard; Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves

doi: 10.1007/s00026-011-0115-4pmid: N/A

We prove a Cauchy identity for free quasi-symmetric functions and apply it to the study of various bases. A free Weyl formula and a generalization of the splitting formula are also discussed.

Huggett, Stephen; Moffatt, Iain

doi: 10.1007/s00026-011-0116-3pmid: N/A

We define 2-decompositions of ribbon graphs, which generalize 2-sums and tensor products of graphs. We give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.

Lozin, Vadim

doi: 10.1007/s00026-011-0117-2pmid: N/A

In the present paper, we identify the first two minimal with respect to set-inclusion hereditary classes of graphs of unbounded clique-width: Bipartite permutation graphs and unit interval graphs.

Rubei, Elena

doi: 10.1007/s00026-011-0118-1pmid: N/A

Let T be a weighted tree with n leaves numbered by the set {1, . . . , n}. Let D i, j (T) be the distance between the leaves i and j. Let $${{D_{i,j,k}(T) = \frac{1}{2}(D_{i,j}(T)+D_{j,k}(T)+D_{i,k}(T))}}$$ . We will call such numbers “triple weights” of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a n-set to be double weights of a tree. By using the same ideas, we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an n-set to be the set of the triple weights of a tree with n leaves. Besides we propose a slight modification of Saitou-Nei’s Neighbour-Joining algorithm to reconstruct trees from the data D i, j .