We give a short proof that a uniform noncrossing partition of the regular $n$-gon weakly converges toward Aldous's Brownian triangulation of the disk, in the sense of the Hausdorff topology. This result was first obtained by Curien & Kortchemski, using a more complicated encoding. Thanks to a result of Marchal on strong convergence of Dyck paths toward the Brownian excursion, we furthermore give an algorithm that allows to recursively construct a sequence of uniform noncrossing partitions for which the previous convergence holds almost surely. In addition, we also treat the case of uniform noncrossing pair partitions of even-sided polygons.
In recent work with Schumann we have proven a conjecture of Naito-Sagaki giving a branching rule for the decomposition of the restriction of an irreducible representation of the special linear Lie algebra to the symplectic Lie algebra, therein embedded as the fixed-point set of the involution obtained by the folding of the corresponding Dyinkin diagram. It provides a new approach to branching rules for non-Levi subalgebras in terms of Littelmann paths. In this paper we motivate this result, provide examples, and give an overview of the combinatorics involved in its proof.
This paper reports the characteristics of and mutual relationships between various combinatorial structures that give rise to certain imprimitive nonsymmetric three-class association schemes. Nontrivial relation graphs of an imprimitive symmetric 2-class association scheme are m ◦ Kr, (the union of m copies of the complete graph on r vertices) and its complement m ◦Kr, (the complete m-partite strongly regular graph) for some positive integers m and r. The set of nontrivial relation graphs of nonsymmetric three-class fission scheme of such a 2-class association scheme contains a complementary pair of oriented graphs of either m◦Kr or m ◦Kr depending on m and r. For suitable parameters m and r, these graphs arise from various combinatorial objects, such as, doubly regular tournaments, normally regular digraphs, skew Hadamard matrices, Cayley graphs of dicyclic groups and certain group rings. The construction and the characteristics of these objects are investigated combinatorially and algebraically, and their mutual relationships are discussed.
A generating set for the wreath product $\ZZ_r \wr S_n$ which leads to a nicely behaved weak order is presented, and properties of the resulting order are studied.
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e. skew Schur functions having Littlewood-Richardson coefficients always equal to $1$ over the full interval.
We give a generalisation of a conjecture by Propp on a summa- tion formula for fully packed loop configurations. The original conjecture states that the number of configurations in which each external edge is connected to its neighbour is equal to the total number of configurations of size one less. This conjecture was later generalised by Zuber to include more types of con- figurations. Our conjecture further refines the counting and provides a general framework for some other summation formulas observed by Zuber. It also im- plies similar summation formulas for half-turn symmetric configurations.
We present a decomposition strategy for c-nets, i. e., root ed 3-connected planar maps. The decomposition yields an algebraic equation for the number of c-nets with a given number of vertices and a given size of the outer face. The decomposition also leads to a deterministic and polynomial time algorithm to sample c-nets uniformly at random. Using rejection sampling, we can also sample isomorphism types of convex polyhedra, i.e., 3-connected planar graphs, uniformly at r andom. R ´ ESUM´ E. Nous proposons une strategie de decomposition pour les cartes pointees 3- connexes (c-reseaux). Cette decomposition permet d'obtenir une ´ eq uation algebrique pour le nombre de c-reseaux suivant le nombre de sommets et la taille de la face exterieure. On en deduit un algorithme de complexite en temps polynomiale pour le tirage aleatoire uniforme des c-reseaux. En utilisant une methoderejet, nous obtenons aussi un algo- rithme de tirage aleatoire uniforme pour les graphes planaires 3-connexes.
Summary. We study several enumerative properties of the set of all binary strings without zigzags, i.e., without substrings equal to 101 or 010. Specifically we give the generating series, a recurrence and two explicit formulas for the number wm,n of these strings with m 1’s and n 0’s and in particular for the numbers wn = wn,n of central strings. We also consider two matrices generated by the numbers wm,n and we prove that one is a Riordan matrix and the other one has a decomposition LTL t where L is a lower triangular matrix and T is a tridiagonal matrix, both with integer entries. Finally, we give a combinatorial interpretation of the strings under consideration as binomial lattice paths without zigzags. Then we consider the more general case of Motzkin, Catalan, and trinomial paths without zigzags.
The present paper gives an introduction to the theory of association schemes, following Bose-Mesner (1959), Biggs (1974), Delsarte (1973), Bannai-Ito (1984) and Brouwer-Cohen-Neumaier (1989). Apart from definitions and many examples, also several proofs and some problems are included. The paragraphs have the following titles: 1. introduction, 2. Distance regular graphs, 3. Minimal idempotents, 4. A-modules, 5. Representations, 6. Root lattices, 7. Generalizations, 8. References.
