The writhe of permutations and random framed knots

We introduce and study the writhe of a permutation, a circular variant of the well‐known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non‐Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied with Hass, Linial, and Nowik (Discrete Comput Geom, 2016). The distribution of the framing in this model is equivalent to the writhe of random permutations. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 121–142, 2017     

How Many Comparisons Does Quicksort Use?

We examine the probability distribution of the number of comparisons needed by the Quicksort sorting algorithm, where probability arises due to the items being in random order. We develop a general class of distributions for the permutation of the items to be sorted which includes the uniform distribution on permutations as a special case. For this general class, the distribution of the number of comparisons is given by the solution of a simple recurrence relation. We provide an exact solution of the recurrence for very small n. We show that the solution can be approximated asymptotically by the solution of a "quadratic" operator equation. We exhibit three numerical solutions to the operator equation for two different distributions on permutations from the general class. We also exhibit numerical solutions for the case in which the algorithm is modified so that arrays are partitioned by the median-of-three selected items rather than by a single selected item.     

Spherically symmetric random permutations

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group S_n and are consistent as n varies. The extreme infinitely spherically symmetric permutation‐valued processes are identified for the Hamming, Kendall‐tau and Cayley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.     

Infinitely Divisible Limit Processes for the Ewens Sampling Formula

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a partial sum process weakly converges in a function space if and only if the corresponding process defined through sums of independent random variables weakly converges. As a consequence of this result, necessary and sufficient conditions for weak convergence to a stable process are established. A counterexample showing that these conditions are not necessary for the one-dimensional convergence is presented. Very few results on the necessity part are known in the literature.     

Applications of conditional power function of two-sample permutation test

Permutation or randomization test is a nonparametric test in which the null distribution (distribution under the null hypothesis of no relationship or no effect) of the test statistic is attained by calculating the values of the test statistic overall permutations (or by considering a large number of random permutation) of the observed dataset. The power of permutation test evaluated based on the observed dataset is called conditional power. In this paper, the conditional power of permutation tests is reviewed. The use of the conditional power function for sample size estimation is investigated. Moreover, reproducibility and generalizability probabilities are defined. The use of these probabilities for sample size adjustment is shown. Finally, an illustration example is used.     

On generation of permutations through decomposition of symmetric groups into cosets

A hardware-oriented algorithm for generating permutations is presented that takes as a theoretic base an iterative decomposition of the symmetric groupS _n into cosets. It generates permutations in a new order. Simple ranking and unranking algorithms are given. The construction of a permutation generator is proposed which contains a cellular permutation network as a main component. The application of the permutation generator for solving a class of combinatorial problems on parallel computers is suggested.     

Generating Functions for Permutations Avoiding a Consecutive Pattern

Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1, . . . , i+j−1 in σ have the same relative order as the elements of τ. We have been able to take advantage of the results of Mendes and Remmel [1] to obtain a generating function for the number of permutations with no τ-matches for several new classes of permutations. These new classes include a large class of permutations which are shuffles of an increasing sequence 1 2 · · · n with an arbitrary permutation σ of the integers {n + 1, . . . , n + m}. Finally we give a formula for the generating function for the number of permutations which have no 1 3 2 4-matches.     

Joint Distributions for Movements of Elements in Sattolo's and the Fisher-Yates Algorithm

Sattolo's algorithm creates a random cyclic permutation by interchanging pairs of elements in an appropriate manner; the Fisher-Yates algorithm produces random (not necessarily cyclic) permutations in a very similar way. The distributions of the movements of the elements in these two algorithms have already been treated quite extensively in past works. In this paper, we are interested in the joint distribution of two elements j and k; we are able to compute the bivariate generating functions explicitly, although it is quite involved. From it, moments and limiting distributions can be deduced. Furthermore, we compute the probability that elements i and j ever change places in both algorithms.     

Stick breaking process generated by virtual permutations with Ewens distribution

Given a sequence x of points in the unit interval, we associate with it a virtual permutation w=w(x) (that is, a sequence w of permutationsw _n such that for all n=1,2,..., w_n−1=w′_n is obtained from w_n by removing the last element n from its cycle). We introduce a detailed version of the well-known stick breaking process generating a random sequence x. It is proved that the associated random virtual permutation w(x) has a Ewens distribution. Up to subsets of zero measure, the space of virtual permutations is identified with the cube [0, 1]^∞. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 162–180.     

Local probabilities and total variation distance for random permutations

We are concerned with permutations taken uniformly at random from the symmetric group. Firstly, we study the probability of a permutation missing short cycles. Secondly, the result is employed to establish a formula for total variation distance between the process of multiplicities of cycle lengths in a random permutation and a process of independent Poisson random variables. We apply an analytic approach originated in number theory (K. Gyory et al. (Eds.) in Number Theory in Progress, 1999).     

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.     

Spatial random permutations with small cycle weights

We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps. We prove the occurrence of infinite macroscopic cycles above a certain critical density.     

Evacuation and a geometric construction for Fibonacci tableaux

Tableaux have long been used to study combinatorial properties of permutations and multiset permutations. Discovered independently by Robinson and Schensted and generalized by Knuth, the Robinson–Schensted correspondence has provided a fundamental tool for relating permutations to tableaux. In 1963, Schützenberger defined a process called evacuation on standard tableaux which gives a relationship between the pairs of tableaux (P,Q) resulting from the Schensted correspondence for a permutation and both the reverse and the complement of that permutation. Viennot gave a geometric construction for the Schensted correspondence and Fomin described a generalization of the correspondence which provides a bijection between permutations and pairs of chains in Young's lattice.In 1975, Stanley defined a Fibonacci lattice and in 1988 he introduced the idea of a differential poset. Roby gave an insertion algorithm, analogous to the Schensted correspondence, for mapping a permutation to a pair of Fibonacci tableaux. The main results of this paper are to give an evacuation algorithm for the Fibonacci tableaux that is analogous to the evacuation algorithm on Young tableaux and to describe a geometric construction for the Fibonacci tableaux that is similar to Viennot's geometric construction for Young tableaux.     

Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation

PS_N is a fast forward permutation if for each m the computational complexity of evaluating P^m(x) is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation in S_N is Θ(N) if one does not use queries of the form P^m(x), but is only Θ(1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from random permutations even when queries of the form P^m(x) are allowed. This is done by introducing an efficient method to sample the cycle structure of a random permutation, which in turn solves an open problem of Naor and Reingold.     

Simple permutations and pattern restricted permutations

A simple permutation is one that does not map any non-trivial interval onto an interval. It is shown that, if the number of simple permutations in a pattern restricted class of permutations is finite, the class has an algebraic generating function and is defined by a finite set of restrictions. Some partial results on classes with an infinite number of simple permutations are given. Examples of results obtainable by the same techniques are given; in particular it is shown that every pattern restricted class properly contained in the 132-avoiding permutations has a rational generating function.     

A new algorithm for generation of permutations

A new algorithm for generating permutations is presented, that generates the next permutation by reversing a certain suffix of its predecessor. The average size of this suffix is less thane 2.8. It is shown how to find the position of a given permutation and how to construct the permutation of a given position, where the position refers to the order in which the permutations are generated, and is also new.On leave from the Department of Computer Science, Technion, Haifa, Israel. This work was supported by NSF grant MCS-8302391.     

The characteristic polynomial of a random permutation matrix at different points

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enable us to study also the wreath product of permutation matrices and diagonal matrices with i.i.d. entries and more general class functions on the symmetric group with a multiplicative structure.     

The peak and descent statistics over ballot permutations

A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. By using a reversal-concatenation map, we (i) give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, (ii) connect this distribution and the joint distribution (pk, des) over ordinary permutations in terms of generating functions, and (iii) confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.