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

On Derivatives and Subpattern Orders of Countable Subshifts

We study the computational and structural aspects of countable two-dimensional SFTs and other subshifts. Our main focus is on the topological derivatives and subpattern posets of these objects, and our main results are constructions of two-dimensional countable subshifts with interesting properties. We present an SFT whose iterated derivatives are maximally complex from the computational point of view, a sofic shift whose subpattern poset contains an infinite descending chain, a family of SFTs whose finite subpattern posets contain arbitrary finite posets, and a natural example of an SFT with infinite Cantor-Bendixon rank.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249