Related papers: Sorting by shuffling methods and a queue
We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square…
This article introduces an adaptive sorting algorithm that can relocate elements accurately by substituting their values into a function which we name it the guessing function. We focus on building this function which is the mapping…
Divide a deck of $kn$ cards into $k$ equal piles and place them from left to right. The standard shuffle $\sigma$ is performed by picking up the top cards one by one from left to right and repeating until all cards have been picked up. For…
We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an…
A mathematical characterization of serially-pruned permutations (SPPs) employed in variable-length permuters and their associated fast pruning algorithms and architectures are proposed. Permuters are used in many signal processing systems…
In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass…
The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice…
In-place associative integer sorting technique was proposed for integer lists which requires only constant amount of additional memory replacing bucket sort, distribution counting sort and address calculation sort family of algorithms. The…
We introduce a novel class of finite difference approximations, termed zigzag schemes, that employ a hybrid stencil that is neither symmetrical, nor fully one-sided. These zigzag schemes often enjoy more permissive stability constraints and…
This paper gives a straightforward self-contained proof of the formula for the variance of the number of comparisons used by the Quicksort sorting algorithm when pivots are chosen uniformly at random. The result has been known for some time…
We present an algorithm, called BiSC, that describes the patterns avoided by a given set of permutations. It automatically conjectures the statements of known theorems such as the descriptions of stack-sortable (Knuth 1975) and…
Pop-stacks are variants of stacks that were introduced by Avis and Newborn in 1981. Coincidentally, a 1982 result of Unger implies that every permutation of length n can be sorted by n-1 passes through a deterministic pop-stack. We give a…
Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the…
Rather than directly considering the queuing delay of data, this memo focuses on reducing the delay that congestion signals experience within a queue management algorithm, which can be greater than the delay that the data itself experiences…
A deck of $n$ cards are shuffled by repeatedly taking off the top card, flipping it with probability $1/2$, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group $B_n$ of…
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.
We study preimages of permutations under the bubblesort operator $\mathbf{B}$. We achieve a description of these preimages much more complete than what is known for the more complicated sorting operators $\mathbf{S}$ (stacksort) and…
Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…
In this expository article, we highlight the direct connection between card shuffling and the functions known as $P$-partitions that come from algebraic combinatorics. While many (but not all) of the results we discuss are known, we give a…