Jean-Marc Fedou

On the tiling by translation problem

On square or hexagonal lattices, tiles or polyominoes are coded by words. The polyominoes that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. We also have a linear algorithm for pseudo-hexagon polyominoes not containing arbitrarily large square factors. The results are extended to more general tiles.

Attribute grammars are useful for combinatorics

Abstract The purpose of this paper is to show the use of attribute grammars in solving some combinatorics problems. For example, we give the generating function for the shape of skew Ferrers diagrams according to the number of cells and the number of columns. This result is new and proves that skew Ferrer diagrams are related to new basic Bessel functions.

New Strahler numbers for rooted plane trees

In this paper, wepresent an extension of Strahler numbers to rooted plane trees. Several asymptotic properties are proved; others are conjectured. We also describe several applications of this extension.

Object grammars and bijections

A new systematic approach for the specification of bijections between sets of combinatorial objects is presented. It is based on the notion of object grammars. Object grammars give recursive descriptions of objects and generalize context-free grammars. The study of a particular substitution in these object grammars confirms once more the key role of Dyck words in the domain of enumerative and bijective combinatorics.

Assessing the quality of multilevel graph clustering

Abstract“Lifting up” a non-hierarchical approach to handle hierarchical clustering by iteratively applying the approach to hierarchically cluster a graph is a popular strategy. However, these lifted iterative strategies cannot reasonably guide the overall nesting process precisely because they fail to evaluate the very hierarchical character of the clustering they produce. In this study, we develop a criterion that can evaluate the quality of the subgraph hierarchy. The multilevel criterion we present and discuss in this paper generalizes a measure designed for a one-level (flat) graph clustering to take nesting of the clusters into account. We borrow ideas from standard techniques in algebraic combinatorics and exploit a variable

Computation and images in combinatorics

q

More Statistics on Permutation Pairs

q to keep track of the depth of clusters at which edges occur. Our multilevel measure relies on a recursive definition involving variable