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We prove that the set of permutations sorted by a stack of depth $t geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a sorting process with two stacks in series, the basis changes from finite to infinite.
We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $kinmathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{dh}mundsson. Our characterisation dema
It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densitie
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of a set of edges meeting every non-empty dicut equa
A general method for constructing sharply $k$-arc-transitive digraphs, i.e. digraphs that are $k$-arc-transitive but not $(k+1)$-arc-transitive, is presented. Using our method it is possible to construct both finite and infinite examples. The infinit