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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.
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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 demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called emph{$2$-avoidance}.
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 densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.
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 equals the maximum number of disjoint dicuts in that digraph. Such sets are called dijoins. Woodall conjectured a dual statement. He asked whether the maximum number of disjoint dijoins in a directed graph equals the minimum size of a non-empty dicut. We study a modification of this question where we restrict our attention to certain classes of non-empty dicuts, i.e. whether for a class $mathfrak{B}$ of dicuts of a directed graph the maximum number of disjoint sets of edges meeting every dicut in $mathfrak{B}$ equals the size of a minimum dicut in $mathfrak{B}$. In particular, we verify this questions for nested classes of finite dicuts, for the class of dicuts of minimum size, and for classes of infinite dibonds, and we investigate how this generalised setting relates to a capacitated version of this question.
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 infinite examples can have one, two or infinitely many ends. Among the one-ended examples there are also digraphs that have polynomial growth.
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