Free sequences

Context-Free Sequences

A sequence w over a finite alphabet A is generated by a uniform automaton if there exists an automaton labelled on {0, . . . , k − 1} for some k > 1 and recognizing for each output a in A the set of positions of a in w expressed in base k. Automatic sequences are generated by finite automata. By considering pushdown automata instead of finite ones, we generate exactly the context-free sequences...

Diamond-free Degree Sequences

While attempting to classify partial linear spaces produced during the execution of an extension of Stinson’s hill-climbing algorithm a new problem arises, that of generating all graphical degree sequences that are diamond-free (i.e. have no diamond as subgraph) and satisfy additional constraints. We formalize this new problem, propose a constraint programming solution and list all satisfying d...

On prefix–free and suffix–free sequences of integers

Binomial regular sequences and free sums

Recently, several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note, we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. We characterize when a binomial regular sequ...

Degree Sequences of F-Free Graphs

Let F be a fixed graph of chromatic number r + 1. We prove that for all large n the degree sequence of any F -free graph of order n is, in a sense, close to being dominated by the degree sequence of some r-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better esti...