Operads from posets and Koszul duality

We introduce a functor ${\sf As}$ from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction ${\sf As}$ provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from ${\sf As}$ are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction ${\sf As}$ restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.Comment: 40 page

Small Superpatterns for Dominance Drawing

We exploit the connection between dominance drawings of directed acyclic graphs and permutations, in both directions, to provide improved bounds on the size of universal point sets for certain types of dominance drawing and on superpatterns for certain natural classes of permutations. In particular we show that there exist universal point sets for dominance drawings of the Hasse diagrams of width-two partial orders of size O(n^{3/2}), universal point sets for dominance drawings of st-outerplanar graphs of size O(n\log n), and universal point sets for dominance drawings of directed trees of size O(n^2). We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}), riffle permutations (321-, 2143-, and 2413-avoiding permutations) have superpatterns of size O(n), and the concatenations of sequences of riffles and their inverses have superpatterns of size O(n\log n). Our analysis includes a calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of the 321-superpattern siz

Logspace computations in graph products

We consider three important and well-studied algorithmic problems in group theory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conjugacy problem. We have a general result for graph products, but even in the special case of a graph group the result is new. Graph groups are closely linked to the theory of Mazurkiewicz traces which form an algebraic model for concurrent processes. Our proofs are combinatorial and based on well-known concepts in trace theory. We also use rewriting techniques over traces. For the group-theoretical part we apply Bass-Serre theory. But as we need explicit formulae and as we design concrete algorithms all our group-theoretical calculations are completely explicit and accessible to non-specialists

The Widths of Strict Outerconfluent Graphs

Strict outerconfluent drawing is a style of graph drawing in which vertices are drawn on the boundary of a disk, adjacencies are indicated by the existence of smooth curves through a system of tracks within the disk, and no two adjacent vertices are connected by more than one of these smooth tracks. We investigate graph width parameters on the graphs that have drawings in this style. We prove that the clique-width of these graphs is unbounded, but their twin-width is bounded.Comment: 15 pages, 2 figure

Application of graph combinatorics to rational identities of type A

To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).Comment: This is the complete version of the submitted fpsac paper (2009