Set Of Permutations Research Articles

Decomposing Sets of Inversions

In this paper we consider the question how the set of inversions of a permutation $\pi \in S_n$ can be partitioned into two subsets, which are themselves inversion sets of permutations in $S_n$. Our method is to study the modular decomposition of the inversion graph of $\pi$. A correspondence to the substitution decomposition of $\pi$ is also given. Moreover, we consider the special case of multiplicative decompositions.

INTERVAL STRUCTURE OF THE PIERI FORMULA FOR GROTHENDIECK POLYNOMIALS

We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way, we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occurring in the result is actually an interval of the Bruhat order.

Permutations and Pairs of Dyck Paths

We define a map v between the symmetric group Sn and the set of pairs of Dyck paths of semilength n. We show that the map v is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.

Sequence Covering Arrays

Sequential processes can encounter faults as a result of improper ordering of subsets of the events. In order to reveal faults caused by the relative ordering of $t$ or fewer of $v$ events, for some fixed $t$, a test suite must provide tests so that every ordering of every set of $t$ or fewer events is exercised. Such a test suite is equivalent to a sequence covering array, a set of permutations on $v$ events for which every subsequence of $t$ or fewer events arises in at least one of the permutations. Equivalently it is a (different) set of permutations, a completely $t$-scrambling set of permutations, in which the images of every set of $t$ chosen events include each of the $t!$ possible “patterns.” In event sequence testing, minimizing the number of permutations used is the principal objective. By developing a connection with covering arrays, lower bounds on this minimum in terms of the minimum number of rows in covering arrays are obtained. An existing bound on the largest $v$ for which the minimum can equal $t!$ is improved. A conditional expectation algorithm is developed to generate sequence covering arrays whose number of permutations never exceeds a specified logarithmic function of $v$ when $t$ is fixed, and this method is shown to operate in polynomial time. A recursive product construction is established when $t=3$ to construct sequence covering arrays on $vw$ events from ones on $v$ and $w$ events. Finally computational results are given for $t \in \{3,4,5\}$ to demonstrate the utility of the conditional expectation algorithm and the product construction.

Cycles and sorting index for matchings and restricted permutations

We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.

Operators of equivalent sorting power and related Wilf-equivalences

We study sorting operators $\textrm{A}$ on permutations that are obtained composing Knuth's stack sorting operator \textrmS and the reverse operator $\textrm{R}$, as many times as desired. For any such operator $\textrm{A}$, we provide a bijection between the set of permutations sorted by $\textrm{S} \circ \textrm{A}$ and the set of those sorted by $\textrm{S} \circ \textrm{R} \circ \textrm{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on an apparently novel bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding families of Wilf-equivalent permutation classes.

The Descriptive Complexity of Series Rearrangements

We consider the descriptive complexity of some subsets of the infinite permutation group \(S_\infty\) which arise naturally from the classical series rearrangement theorems of Riemann, Levy, and Steinitz. In particular, given some fixed conditionally convergent series of vectors in Euclidean space \(\mathbb{R}^d\), we study the set of permutations which make the series diverge, as well as the set of permutations which make the series diverge properly. We show that both collections are \(\boldsymbol\Sigma^0_3\)-complete in \(S_\infty\), regardless of the particular choice of series.

Maximum Size of Reverse-Free Sets of Permutations

Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a reverse-free set of words from [n]^k where no letter repeats within a word. We show the following lower and upper bounds in the case n >= k: F(n,k) \in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.

Some simple varieties of trees arising in permutation analysis

After extending classical results on simple varieties of trees to trees counted by their number of leaves, we describe a filtration of the set of permutations based on their strong interval trees. For each subclass we provide asymptotic formulas for number of trees (by leaves), average number of nodes of fixed arity, average subtree size sum, and average number of internal nodes. The filtration is motivated by genome comparison of related species. Nous commençons par étendre les résultats classiques sur les variétés simples d'arbres aux arbres comptés selon leur nombre de feuilles, puis nous décrivons une filtration de l'ensemble des permutations qui repose sur leurs arbres des intervalles communs. Pour toute sous-classe, nous donnons des formules asymptotiques pour le nombre d'arbres (comptés selon les feuilles), le nombre moyen de nœuds d'arité fixée, la moyenne de la somme des tailles des sous-arbres, et le nombre moyen de nœuds internes. Cette filtration est motivée par des problématiques de comparaison de génomes.

GROUP OF POLYNOMIAL PERMUTATIONS OF ℤpr

Abstract. The set of all polynomial permutations of Z p r forms agroup. We investigate the structure of the group and some relatedgroups, and completely determine the structure of the group of allpolynomial permutations of Z p 2 . 1. IntroductionLet p r be a prime power. If a polynomial over the Galois ring Z p r induces a permutation of Z p r , then it is called a permutation polynomial.For r= 1, it is well-known that every permutation of the eld Z p isinduced by a polynomial [4]. On the other hand, for r>1, not everypermutation of Z p r is induced by a polynomial. Hence the notion of apolynomial permutation, that is, permutation induced by a polynomialis meaningful in this case.It is easy to see that the set of all polynomial permutations of Z p r isa group. Indeed the set of all polynomial permutations of Z p r is clearlyclosed under composition and is a nite subset of the symmetric groupof Z p r , and hence forms a subgroup. We investigate the structure ofthis group and related groups. In particular, we completely determinethe structure of the group of all polynomial permutations of Z

Schur-convexity of 2nd order, certain subclass of multivariate arrangement increasing functions with applications in statistics

It is shown that starting with a certain meaningful problem of the type “ranking of populations”, a need arises to employ functions which we call “Schur-convex of 2nd order with respect to two variables”. These functions L(v1,v2,v3,…,vn) are symmetric, and they are characterized in essence by the relation Lv12″−2Lv1v2″+Lv22″≥0. It is shown that this subclass of Schur-convex functions is closely related to a certain subclass of multivariate arrangement increasing functions introduced by Boland and Proschan [P.J. Boland, F. Proschan, Multivariate arrangement increasing functions with applications in probability and statistics, J. Multivariate Anal. (1988) 25 286–298]. This relation allows us to solve a series of statistical problems concerning maximization of the goal function with respect to the risk criterion on the set of permutations of the function’s arguments.For families of distributions with location parameter, it is shown that the ranking of parameters by sample means is expedient in the case of families having no sufficient statistic. Furthermore, mean losses (the risk) arising because of making wrong decisions decreases with growth of the size of the sample if densities are symmetric and log- concave. Examples of applications of these results are given.

A categorical approach to loops, neardomains and nearfields

In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups. The other categories are also shown to be equivalent with categories whose objects are sets of permutations with suitable extra properties. Up to now the equivalence between neardomains and sharply 2-transitive groups was only known when both categories were equipped with the obvious isomorphisms as morphisms. We thank Hubert Kiechle for this observation.

On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan. In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$. The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$. We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result. Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index. In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$. We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$. In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

Signatures of Strings

The parity of a permutation π can be defined as the parity of the number of inversions in π. The signature e(π) of π is an encoding of the parity in a multiplicative group of order 2: e(π) = (−1)inv(π). It is also well known that half of the permutations of a finite set are even and half are odd. In this paper, we explore the natural notion of parity for larger moduli; that is, we define the m-signature of a permutation π to be the number of inversions of π, reduced modulo m. Unlike with the 2-signatures of permutations of sets, the m-signatures of the permutations of a multiset are not typically equi-distributed among the modulo m residue classes, though the distribution is close to uniform. We present a recursive method of calculating the distribution of m-signatures of permutations of a multiset, develop properties of this distribution, and present sufficient conditions for the distribution to be uniform.

Inapproximability Results for the Weight Problems of Subgroup Permutation Codes

A subgroup permutation code is a set of permutations on n symbols with the property that its elements are closed under the operation of composition. In this paper, we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well-known metrics. Based on previous works, we prove that under Hamming, Lee, Cayley, Kendall's tau, Ulam's, and lp distance metrics, 1) there is no polynomial-time 2log1-en-approximation algorithm for the minimum weight problem for any constant e >; 0 unless NP ⊆ DTIME(2polylog(n)) (quasi-polynomial time), and 2) there is no polynomial-time r-approximation algorithm for the minimum weight problem for any constant r >; 1 unless P = NP. Under l∞-metric, we prove that it is NP-hard to approximate the minimum weight problem within factor 2-e for any constant e >; 0. We also prove that for any constant e >; 0, it is NP-hard to approximate the maximum weight within p √{[ 3/ 2]}-e under lp distance metric, and within [ 3/ 2]-e under Hamming, Lee, Cayley, Kendall's tau, and Ulam's distance metrics.

Statistics-preserving bijections between classical and cyclic permutations

Recently, Elizalde (2011) [2] has presented a bijection between the set Cn+1 of cyclic permutations on {1,2,…,n+1} and the set of permutations on {1,2,…,n} that preserves the descent set of the first n entries and the set of weak excedances. In this paper, we construct a bijection from Cn+1 to Sn that preserves the weak excedance set and that transfers quasi-fixed points into fixed points and left-to-right maxima into themselves. This induces a bijection from the set Dn of derangements to the set Cn+1q of cycles without quasi-fixed points that preserves the weak excedance set. Moreover, we exhibit a kind of discrete continuity between Cn+1 and Sn that preserves at each step the set of weak excedances. Finally, some consequences and open problems are presented.

Statistical inference and multiple testing correction in classification-based multi-voxel pattern analysis (MVPA): Random permutations and cluster size control

An ever-increasing number of functional magnetic resonance imaging (fMRI) studies are now using information-based multi-voxel pattern analysis (MVPA) techniques to decode mental states. In doing so, they achieve a significantly greater sensitivity compared to when they use univariate frameworks. However, the new brain-decoding methods have also posed new challenges for analysis and statistical inference on the group level. We discuss why the usual procedure of performing t-tests on accuracy maps across subjects in order to produce a group statistic is inappropriate. We propose a solution to this problem for local MVPA approaches, which achieves higher sensitivity than other procedures. Our method uses random permutation tests on the single-subject level, and then combines the results on the group level with a bootstrap method. To preserve the spatial dependency induced by local MVPA methods, we generate a random permutation set and keep it fixed across all locations. This enables us to later apply a cluster size control for the multiple testing problem. More specifically, we explicitly compute the distribution of cluster sizes and use this to determine the p-values for each cluster. Using a volumetric searchlight decoding procedure, we demonstrate the validity and sensitivity of our approach using both simulated and real fMRI data sets. In comparison to the standard t-test procedure implemented in SPM8, our results showed a higher sensitivity. We discuss the theoretical applicability and the practical advantages of our approach, and outline its generalization to other local MVPA methods, such as surface decoding techniques.

A $$q$$ -analog of the hyperharmonic numbers

Recently, the $$q$$ -analog of the harmonic numbers obtained by replacing each positive integer $$n$$ with $$n_q$$ has been shown to satisfy congruences which generalize Wolstenholme’s theorem. Here, we wish to consider further algebraic properties of these numbers. Recall that the $$r$$ -harmonic, or hyperharmonic, numbers arise by taking repeated partial sums of harmonic numbers. In this paper, we introduce and study properties of a $$q$$ -analog of the $$r$$ -harmonic numbers, which reduces to the aforementioned $$q$$ -harmonic numbers when $$r=1$$ . It is defined in terms of a statistic on the set of permutations of length $$n$$ in which the elements $$1,2,\ldots ,r$$ belong to distinct cycles (which is enumerated by the $$r$$ -Stirling number of the first kind).

Uncertainty measure for the Bellman–Johnson problem with interval processing times

Time-optimal processing of jobs with identical processing routes by two different machines is considered. The processing time of a job becomes known only after the completion of the job (it is only known that this time belongs to a given interval). For such a problem, an uncertainty measure is introduced. This measure is based on the cardinality of the minimal dominant set of job permutations. A uniqueness criterion is obtained for the minimal dominant set.

Two Permutation Classes related to the Bubble Sort Operator

We introduce the Dual Bubble Sort operator $\hat{B}$ (a sorting algorithm such that, if $\sigma=\alpha\,1\,\beta$ is a permutation, then $\hat{B}(\sigma)=1\,\alpha\,\hat{B} (\beta)$) and consider the set of permutations sorted by the composition $\hat{B}B$, where $B$ is the classical Bubble Sort operator. We show that this set is a permutation class and we determine the generating function of the descent and fixed point distributions over this class. Afterwards, we characterize the same distributions over the set of permutations that are sorted by both $\hat{B}^2$ and $B^2$.