This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By enumerating some of the remaining classes we bring that upper-bound further down to 56. Finally, we conjecture that the actual number of Wilf-classes of mesh patterns of length 2 is 46.

A graph <i>G</i>=(<i>V,E</i>) is representable if there exists a word <i>W</i> over the alphabet <i>V</i> such that letters <i>x</i> and <i>y</i> alternate in <i>W</i> if and only if (<i>x</i> ,<i>y</i>) is in <i>E</i> for each <i>x</i> not equal to <i>y</i> . The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for <i>n</i> greater than 3, the line graph of an <i>n</i>-wheel is non-representable. This not only provides a new construction of non-repre- sentable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for <i>n</i> greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph <i>G</i> which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of <i>G k</i>-times is guaranteed to be non-representable for <i>k</i> greater than 3

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International audience We prove new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of factorial and Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical patterns where conditions are placed on the location of an occurrence in a permutation, as well as on the values in the occurrence. This clarifies what happens when the requirement of smoothness is weakened to factoriality and further to Gorensteinness, extending work of Bousquet-Mélou and Butler (2007), and Woo and Yong (2006). We also prove results that translate some known patterns in the literature into bivincular patterns. Nous démontrons de nouveaux liens entre les motifs de permutation et les singularités des variétés de Schubert, par la méthode de donner une nouvelle caractérisation des variétés factorielles et de Gorenstein par rapport à les motifs bivinculaires. Ces motifs sont généralisations des motifs classiques où des conditions se posent sur la position d'une occurrence dans une permutation, aussi bien que sur les valeurs qui se présentent dans l'occurrence. Ceci éclaircit les phénomènes où la condition de nonsingularité s'affaiblit á factorialité et même à Gorensteinité, et augmente les travaux de Bousquet-Mélou et Butler (2007), et de Woo et Yong (2006). Nous démontrons également des résultats qui traduisent quelques motifs connus en la littérature en motifs bivinculaires.