Methods of computing deque sortable permutations given complete and incomplete information

The problem of determining which permutations can be sorted using certain switchyard networks dates back to Knuth in 1968. In this work, we are interested in permutations which are sortable on a double-ended queue (called a deque), or on two parallel stacks. In 1982, Rosenstiehl and Tarjan presented an O(n) algorithm for testing whether a given permutation was sortable on parallel stacks. In the same paper, they also presented a modification giving O(n) test for sortability on a deque. We demonstrate a slight error in the version of their algorithm for testing deque sortability, and present a fix for this problem. The general enumeration problem for both of these classes of permutations remains unsolved. What is known is that the growth rate of both classes is approximately Theta(8^n), so computing the number of sortable permutations of length n, even for small values of n, is difficult to do using any method that must evaluate each sortable permutation individually. As far as we know, the number of deque sortable permutations was known only up to n=14. This was computed using algorithms which effectively generate all sortable permutations. By using the symmetries inherent in the execution of Tarjan's algorithm, we have developed a new dynamic programming algorithm which can count the number of sortable permutations in both classes in O(n^5 2^n) time, allowing the calculation of the number of deque and parallel stack sortable permutation for much higher values of n than was previously possible.Comment: dartmouth senior honors thesis advised by Peter Doyle and Scot Drysdale 45 pages, 9 figure

Two Vignettes On Full Rook Placements

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ru\v{s}kuc.Comment: 9 pages, 4 figure

Sorting by shuffling methods and a queue

We consider sorting by a queue that can apply a permutation from a given set over its content. This gives us a sorting device $\mathbb{Q}_{\Sigma}$ corresponding to any shuffling method $\Sigma$ since every such method is associated with a set of permutations. Two variations of these devices are considered - $\mathbb{Q}_{\Sigma}^{\prime}$ and $\mathbb{Q}_{\Sigma}^{\text{pop}}$. These require the entire content of the device to be unloaded after a permutation is applied or unloaded by each pop operation, respectively. First, we show that sorting by a deque is equivalent to sorting by a queue that can reverse its content. Next, we focus on sorting by cuts. We prove that the set of permutations that one can sort by using $\mathbb{Q}_{\text{cuts}}^{\prime}$ is the set of the $321$-avoiding separable permutations. We give lower and upper bounds to the maximum number of times the device must be used to sort a permutation. Furthermore, we give a formula for the number of $n$-permutations, $p_{n}(\mathbb{Q}_{\Sigma}^{\prime})$, that one can sort by using $\mathbb{Q}_{\Sigma}^{\prime}$, for any shuffling method $\Sigma$, such that the permutations associated with it are irreducible. Next, we prove a generalization of the fact that $\mathbb{Q}_{\text{cuts}}^{\text{pop}}$ can sort all permutations. We also show that $p_{n}(\mathbb{Q}_{\Sigma}^{\text{pop}})$ is given by the odd indexed Fibonacci numbers $F_{2n-1}$, for any shuffling method $\Sigma$ having a specific back-front property. The rest of the work is dedicated to a surprising conjecture inspired by Diaconis and Graham which states that one can sort the same number of permutations of any given size by using the devices $\mathbb{Q}_{\text{In-sh}}^{\text{pop}}$ and $\mathbb{Q}_{\text{Monge}}^{\text{pop}}$, corresponding to the popular In-shuffle and Monge shuffling methods.Comment: 29 pages, 7 figure