KAIST Discrete Math Seminar

For each composition we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition . Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module where is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homology.

A hypergraph on V is a collection E of a subsets of a ground set V. A transversal in a hypergraph H=(V,E) is a subset T of V that intersects every set in E. We are interested in an output-polynomial algorithm for listing the set (inclusion wise) minimal transversals in a given hypergraph (known as Hypergraph Dualization or Transversal Problem). An enumeration algorithm for a set C is an algorithm that lists the elements of C without repetitions; it is said output-polynomial if it can enumerate the set C in time polynomial in the size of C and the input. The Transversal problem is a fifty-year open problem and until now not so many tractable cases are known and has deep connections with several areas of computer science: dualization of monotone functions, data mining, artificial intelligence, etc.</p>

In this talk I will review some known results. In particular I will show that the Transversal problem is polynomially reduced to the enumeration of minimal dominating sets in co-bipartite graphs. A dominating set in a graph is a subset of vertices that intersect the closed neighborhood of every vertex. This interesting connection, we hope, will help in solving the Transversal problem by bringing structural graph theory into this area. I will also review new graph classes where we obtain polynomial delay algorithm for listing the minimal dominating sets. The talk will emphasize on the known techniques rather than a listing of known tractable cases.

Consider a polytope P with integer vertices, then one can define its Ehrhart polynomial f(t) by counting integer points in t.P. After a change of basis, it becomes a polynomial with positive integer coefficients, called the h*-polynomial. It is then a problem to find the combinatorial meaning of these coefficients for special polytopes. For example, the n-dimensional hypercube gives the n-th Eulerian polynomial, counting descents in permutations. The goal of this work is to refine this result by considering slices of hypercube and considering descents and excedences in permutations, that are two different Eulerian statistics.</p>

is the rectangle of size in the square lattice. denotes the cardinality of multiple self-avoiding polygons in . </p>

In this talk, we construct an algorithm producing the precise value of for positive integers m,n that uses recurrence relations of state matrices which turn out to be remarkably efficient to count such polygons. $$ p_{m \times n} = \mbox{(1,1)-entry of the matrix } (X_m)^n -1$$ where the matrix is defined by $$ X_{k+1} = \left( \begin{array}{cc} X_k & O_k \\ O_k & X_k \end{array}\right) \ \ \mbox{and} \ \ \ O_{k+1} = \left( \begin{array}{cc} O_k & X_k \\ X_k & 0 \end{array} \right) $$ for , with matrices and .

Let A and B be finite nonempty subsets of a multiplicative group G, and consider the product set AB = { ab | a in A and b in B }. When |G| is prime, a famous theorem of Cauchy and Davenport asserts that |AB| is at least the minimum of {|G|, |A| + |B| – 1}. This lower bound was refined by Vosper, who characterized all pairs (A,B) in such a group for which |AB| < |A| + |B|. Kneser generalized the Cauchy-Davenport theorem by providing a natural lower bound on |AB| which holds in every abelian group. Shortly afterward, Kemperman determined the structure of those pairs (A,B) with |AB| < |A| + |B| in abelian groups. Here we present a further generalization of these results to arbitrary groups. Namely we generalize Kneser’s Theorem, and we determine the structure of those pairs with |AB| < |A| + |B| in arbitrary groups.