In an exercise in the first volume of his famous series of books, Knuth considered sorting permutations by passing them through a stack. Many variations of this exercise have since been considered, including allowing multiple passes through the stack and using different data structures. We are concerned with a variation using pop-stacks that was introduced by Avis and Newborn in 1981. Let P k (x) be the generating function for the permutations sortable by k passes through a pop-stack. The generating function P 2 (x) was recently given by Pudwell and Smith (the case k = 1 being trivial). We show that P k (x) is rational for any k. Moreover, we give an algorithm to derive P k (x), and using it we determine the generating functions P k (x) for k ≤ 6.

No abstract

Permutations in the image of the pop-stack operator are said to be pop-stacked. We give a polynomial-time algorithm to count pop-stacked permutations up to a fixed length and we use it to compute the first 1000 terms of the corresponding counting sequence. Only the first 16 terms had previously been computed. With the 1000 terms we prove some negative results concerning the nature of the generating function for pop-stacked permutations. We also predict the asymptotic behavior of the counting sequence using differential approximation.

We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an enumeration. The algorithm is successful on different inputs than other algorithms and can succeed with any polynomial permutation class. We apply it to every non-polynomial permutation class avoiding a set of length four patterns. The structures found by the algorithm can sometimes allow an enumeration of the permutation class with respect to permutation statistics, as well as choosing a permutation uniformly at random from the permutation class. We sketch a new algorithm formalizing the human verification of the conjectured covers.

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge weights. Finally we give fixed-parameter algorithms for weighted improper coloring both when parameterized by treewidth and maximum degree and when parameterized by treewidth and precision of edge weights. In particular, we obtain a linear-time algorithm for weighted improper coloring of interval graphs of bounded degree.

No abstract

No abstract

No abstract