Restricted Permutation Research Articles

Burstein’s permutation conjecture, Hong and Li’s inversion sequence conjecture and restricted Eulerian distributions

AbstractRecently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation of Hong and Li’s conjecture and Burstein’s first conjecture but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.

Testing independence of bivariate censored data using random walk on restricted permutation graph

In this paper, we propose a procedure to test the independence of bivariate censored data, which is generic and applicable to any censoring types in the literature. To test the hypothesis, we consider a rank-based statistic, Kendall’s tau statistic. The censored data defines a restricted permutation space of all possible ranks of the observations. We propose the statistic, the average of Kendall’s tau over the ranks in the restricted permutation space. To evaluate the statistic and its reference distribution, we develop a Markov chain Monte Carlo (MCMC) procedure to obtain uniform samples on the restricted permutation space and numerically approximate the null distribution of the averaged Kendall’s tau. We numerically compare the power of our procedure to existing state of the art procedures in the literature under various censoring types. We apply the procedure to three real data examples with different censoring types, and compare the results with those by existing methods.

Constraining PERMANOVA and LDM to within-set comparisons by projection improves the efficiency of analyses of matched sets of microbiome data

BackgroundMatched-set data arise frequently in microbiome studies. For example, we may collect pre- and post-treatment samples from a set of individuals, or use important confounding variables to match data from case participants to one or more control participants. Thus, there is a need for statistical methods for data comprised of matched sets, to test hypotheses against traits of interest (e.g., clinical outcomes or environmental factors) at the community level and/or the operational taxonomic unit (OTU) level. Optimally, these methods should accommodate complex data such as those with unequal sample sizes across sets, confounders varying within sets, and continuous traits of interest.MethodsPERMANOVA is a commonly used distance-based method for testing hypotheses at the community level. We have also developed the linear decomposition model (LDM) that unifies the community-level and OTU-level tests into one framework. Here we present a new strategy that can be used with both PERMANOVA and the LDM for analyzing matched-set data. We propose to include an indicator variable for each set as covariates, so as to constrain comparisons between samples within a set, and also permute traits within each set, which can account for exchangeable sample correlations. The flexible nature of PERMANOVA and the LDM allows discrete or continuous traits or interactions to be tested, within-set confounders to be adjusted, and unbalanced data to be fully exploited.ResultsOur simulations indicate that our proposed strategy outperformed alternative strategies, including the commonly used one that utilizes restricted permutation only, in a wide range of scenarios. Using simulation, we also explored optimal designs for matched-set studies. The flexibility of PERMANOVA and the LDM for a variety of matched-set microbiome data is illustrated by the analysis of data from two real studies.ConclusionsIncluding set indicator variables and permuting within sets when analyzing matched-set data with PERMANOVA or the LDM is a strategy that performs well and is capable of handling the complex data structures that frequently occur in microbiome studies.3i1UxvTTxBo8cNUR6xzMG1Video

Permutations with restricted movement

A restricted permutation of a locally finite directed graph $ G = (V, E) $ is a vertex permutation $ \pi: V\to V $ for which $ (v, \pi(v))\in E $, for any vertex $ v\in V $. The set of such permutations, denoted by $ \Omega(G) $, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [18] of restricted $ {\mathbb Z}^d $ permutations, in which $ \Omega(G) $ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted $ {\mathbb Z}^d $-permutations. We discuss the global and local admissibility of patterns, in the context of restricted $ {\mathbb Z}^d $-permutations. Finally, we review the related models of injective and surjective restricted functions.

Changes in subarctic vegetation after one century of land use and climate change

AbstractQuestionsHave species richness and composition in subarctic vegetation changed over the past ca. 90 years? Are compositional shifts linked to changes in land management or climate? Are observed changes associated with vegetation type, life form, or habitat preference?LocationRybachy and Sredny Peninsulas, NW Russia.MethodsWe resurveyed vegetation ca. 90 years after the first sampling in 1927–1930 to study changes in species richness, abundance and composition. Because of missing plot‐related environmental measurements we used a weighted averaging approach calculating relative changes in species‐specific optimum values for different environmental gradients represented by species indicator values to identify compositional change in relation to the environment. Changes in species composition were visualised using detrended correspondence analyses. Significances of observed changes in species richness and frequency were evaluated using restricted permutation tests. A χ2 test was used to test if observed changes in abundances were related to species’ life form and habitat preferences.ResultsSpecies composition has changed significantly over the past ca. 90 years, as indicated by significant changes in species’ frequencies and values of optima for the environmental gradients temperature, moisture, nutrients and light. Species richness decreased significantly, in particular in nitrophilous and wet growing vegetation. Species typical for grazed grasslands and meadows and species of wet habitats became less abundant, while dwarf shrubs and forest species increased.ConclusionsLand abandonment, in combination with climate change, is likely to have caused the observed changes in the subarctic vegetation of NW Russia. Shifts in the species dominance ratios and interspecific competition (e.g. for reallocated nutrients) after land abandonment may have been promoted by the subsequent change towards a warmer climate, facilitating the regrowth of previously open meadows with grazing‐intolerant tall herbs, forest herbs and dwarf shrubs. This study illustrates clearly the long‐term effects of land‐use change, the consequences of which are still visible even after almost one century in the subarctic.

Compositional inverses of permutation polynomials of the form xrh(xs) over finite fields

The study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G. L. Mullen (1991), as well as the potential applications of these permutation polynomials (Dillon 1974, Khachatrian and Kyureghyan, Discrete Appl. Math. 216, 622–626 2017, Lidl 1985, Lidl and Muller 1984, Rivest et al., ACM Commun. Comput. Algebra. 1978, 120–126 1976, Schwenk and Huber, Electron. Lett. 34, 759–760 1998). It is well known that every permutation polynomial over a finite field $\mathbb {F}_{q}$ can be reduced to a permutation polynomial of the form xrh(xs) with s∣(q − 1) and $h(x) \in \mathbb {F}_{q}[x]$ (Akbary et al., Finite Fields Appl. 15(2), 195–206 2009, Wang, Finite Fields Appl. 22, 57–69 2013). Recently, several explicit classes of permutation polynomials of the form xrh(xs) over ${\mathbb F}_{q}$ have been constructed. However, all the known methods to compute the compositional inverses of permutation polynomials of this form seem to be inadequately explicit, which could be a hurdle to potential applications. In this paper, for any prime power q, we introduce a new approach to explicitly compute the compositional inverse of a permutation polynomial of the form xrh(xs) over ${\mathbb F}_{q}$ , where s∣(q − 1) and $\gcd (r,q-1)= 1$ . The main idea relies on transforming the problem of computing the compositional inverses of permutation polynomials over ${\mathbb F}_{q}$ into computing the compositional inverses of two restricted permutation mappings, where one of them is a monomial over $\mathbb {F}_{q}$ and the other is the polynomial xrh(x)s over a particular subgroup of $\mathbb {F}_{q}^{*}$ with order (q − 1)/s. This is a multiplicative analog of Tuxanidy and Wang (Finite Fields Appl. 28, 244–281 2014), Wu and Liu (Finite Fields Appl. 24, 136–147 2013). We demonstrate that the inverses of these two restricted permutations can be explicitly obtained in many cases. As consequences, many explicit compositional inverses of permutation polynomials given in Zieve (Proc. Am. Math. Soc. 137, 2209–2216 2009), Zieve (arXiv: 1310.0776 , 2013), Zieve (arXiv: 1312.1325v3 , 2013) are obtained using this method.

From Grassmann Necklaces to Restricted Permutations and Back Again

We study the commutative algebras ZJK appearing in Brown and Goodearl’s extension of the \(\mathcal {H}\)-stratification framework, and show that if A is the single parameter quantized coordinate ring of Mm,n, GLn or SLn, then the algebras ZJK can always be constructed in terms of centres of localizations. The main purpose of the ZJK is to study the structure of the topological space spec(A), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real m × n matrices in terms of its restricted permutation.

Large climate change, large effect? Vegetation changes over the past century in the European High Arctic

AbstractQuestionsTerrestrial vegetation is assumed to be highly vulnerable to climate change in the Arctic, which has experienced the highest rates of temperature increase globally over the past decades. However, long‐term studies at the community scale are rare, particularly for the European High Arctic. We ask: (1) has Arctic vegetation changed over the past century; (2) can observed changes be related to recent climate change; and (3) do observed changes in vegetation prevail for specific species or species functional groups?LocationThe Svalbard Archipelago, High Arctic Norway.MethodsIn 2009 we resurveyed three plant sociological studies up to 85 yrs after the first surveys. Vegetation data were recorded from 1‐m2 non‐permanently marked plots, and soil pH was measured. We applied restricted permutation tests to evaluate whether observed changes in vegetation (species richness, occurrence frequency, coverage) and in soil pH were significant and independent of plant community type. We compared species co‐occurrences over time to study changes in species composition and to identify the species most likely causing these changes.ResultsTotal number of species and average number of species per plot has not changed, and are 75 and nine, respectively. Occurrence frequency changed significantly for 34% of the 64 species tested (11 species increased, 11 decreased). Species cover decreased significantly for 27% of the species and increased for one species. Observed changes in frequency and cover are not significantly related to species’ functional groups or species affiliated to moist or dry habitats. Polygonum viviparum, Saxifraga cernua and Alopecurus borealis have changed most in co‐occurring with other species. Soil pH increased significantly from 6.4 to 6.8.ConclusionsOur results suggest that the strong climatic warming in the High Arctic over the past decades has contributed to significant changes in the vegetation studied on Svalbard over the past 85 yrs. Internal community restructuring and the overall stability in species richness may be explained by time‐delayed responses of well‐established communities, and/or by a limited size of the species pool in the area.

A new error measure for forecasts of household-level, high resolution electrical energy consumption

As low carbon technologies become more pervasive, distribution network operators are looking to support the expected changes in the demands on the low voltage networks through the smarter control of storage devices. Accurate forecasts of demand at the individual household-level, or of small aggregations of households, can improve the peak demand reduction brought about through such devices by helping to plan the most appropriate charging and discharging cycles. However, before such methods can be developed, validation measures which can assess the accuracy and usefulness of forecasts of the volatile and noisy household-level demand are required. In this paper we introduce a new forecast verification error measure that reduces the so-called “double penalty” effect, incurred by forecasts whose features are displaced in space or time, compared to traditional point-wise metrics, such as the Mean Absolute Error, and p-norms in general. The measure that we propose is based on finding a restricted permutation of the original forecast that minimises the point-wise error, according to a given metric. We illustrate the advantages of our error measure using half-hourly domestic household electrical energy usage data recorded by smart meters, and discuss the effect of the permutation restriction.

A novel complex community network division algorithm with multi-gene families encoding

To overcome the drawbacks of traditional complex community network division algorithms, such as low accuracy, high time complexity and easily being trapped into local optimum, a novel complex community network division algorithm is proposed by using Multi-Gene Families MGF. The proposed approach first encodes the node ID and the community type into two different MGFs, respectively, and then encodes the relationship of the two MGFs into the chromosome through a mapping function. Moreover, in order to prevent premature and speed up convergence, the elite migration strategy is utilised throughout the evolution process, such as gene selection, chromosome crossover, chromosome inversion, restricted permutation. The experiments and analyses show that our approach is better than the traditional evolutionary algorithm in terms of efficiency and accuracy.

Changes in arctic vegetation on Jan Mayen Island over 19 and 80 years

AbstractQuestionsCan recent vegetation changes on an isolated, grazer‐free island be explained by recent climate change? Are observed changes consistent when focusing on two different time scales?LocationJan Mayen, an arctic volcanic island in the North Atlantic Ocean.MethodsWe re‐surveyed two botanical studies conducted 19 and 80 yr earlier to explore changes in species frequency, cover and co‐occurrence with other species. The observed changes were statistically evaluated using restricted permutation tests and were compared for the two time scales considered using Pearson correlation tests.ResultsTotal number of species did not significantly change over the two time periods considered. One species (Botrychium lunaria) was found new to the island. The dwarf‐shrub Salix herbacea and several graminoids increased in frequency or cover, or both, whereas species linked to snowbeds (e.g. Saxifraga spp., Oxyria digyna, Cerastium cerastoides) decreased. Changes over 19 yr were significantly correlated with 80‐yr changes considering species frequency, but not when comparing changes in cover and species co‐occurrences. Observed changes were more pronounced in the 80‐yr comparison.ConclusionsOur findings from the virtually grazer‐free island of Jan Mayen are in line with other studies on short‐ and long‐term vegetation changes in the Arctic and confirm that indirect effects of climate change (e.g. longer growing season, altered soil moisture conditions, increased nutrient availability) may be the main driver of the observed changes in arctic vegetation composition. However, whereas our study found the main trend to be similar over both time scales considered, discrepancies in the trends of some species suggest that long‐term changes are only partly predictable from short‐term studies.

Enumeration of Restricted Permutation Triples

The counting problem is investigated for the permutation triples of the first $n$ natural numbers with exactly $k$ occurrences of simultaneous "rises". Their recurrence relations and bivariate generating functions are established.

Properties of a randomization test for multifactor comparisons of groups

Multivariate hypothesis testing in studies of vegetation is likely to be hindered by unrealistic assumptions when based on conventional statistical methods. This can be overcome by randomization tests. In this paper, the accuracy and power of a MANOVA randomization test are evaluated for one and two factors with interaction with simulated data from three distributions. The randomization test is based on the partitioning of sum of squares computed from Euclidean distances. In one-factor designs, sample size and variance inequality were evaluated. The results showed a high level of accuracy. The power curve was higher with normal distribution, lower with uniform, intermediate with lognormal and was sensitive to variance inequality. In two-factor designs, three methods of permutations and two statistics were compared. The results showed that permutation of the residuals with F pseudo is accurate and can give good power for testing the interaction and restricted permutation for testing main factors.

Permutation Tests for Studying Classifier Performance

We explore the framework of permutation-based p-values for assessing the performance of classifiers. In this paper we study two simple permutation tests. The first test assess whether the classifier has found a real class structure in the data; the corresponding null distribution is estimated by permuting the labels in the data. This test has been used extensively in classification problems in computational biology. The second test studies whether the classifier is exploiting the dependency between the features in classification; the corresponding null distribution is estimated by permuting the features within classes, inspired by restricted randomization techniques traditionally used in statistics. This new test can serve to identify descriptive features which can be valuable information in improving the classifier performance. We study the properties of these tests and present an extensive empirical evaluation on real and synthetic data. Our analysis shows that studying the classifier performance via permutation tests is effective. In particular, the restricted permutation test clearly reveals whether the classifier exploits the interdependency between the features in the data.

Inference after variable selection using restricted permutation methods

When confronted with multiple covariates and a response variable, analysts sometimes apply a variable-selection algorithm to the covariate-response data to identify a subset of covariates potentially associated with the response, and then wish to make inferences about parameters in a model for the marginal association between the selected covariates and the response. If an independent data set were available, the parameters of interest could be estimated by using standard inference methods to fit the postulated marginal model to the independent data set. However, when applied to the same data set used by the variable selector, standard ("naive") methods can lead to distorted inferences. The authors develop testing and interval estimation methods for parameters reflecting the marginal association between the selected covariates and response variable, based on the same data set used for variable selection. They provide theoretical justification for the proposed methods, present results to guide their implementation, and use simulations to assess and compare their performance to a sample-splitting approach. The methods are illustrated with data from a recent AIDS study.

Exclusion Principles as Restricted Permutation Symmetries

We give a derivation of exclusion principles for the elementary particles of the standard model, using simple mathematical principles arising from a set theory of identical particles. We apply the theory of permutation group actions, stating some theorems which are proven elsewhere, and interpreting the results as a heuristic derivation of Pauli's Exclusion Principle (PEP) which dictates the formation of elements in the periodic table and the stability of matter, and also a derivation of quark confinement. We arrive at these properties by using a symmetry property of collections of the particles themselves as compared for example to the symmetry property of their wave function under interchange of two particles.

The rook numbers of Ferrers boards and the related restricted permutation numbers

Abstract The number of ways of placing k non-taking rooks on a Ferrers board Bn is obtained as a finite difference of order n−k of a polynomial of order n. Then the number of permutations of the set {1,2,…,n} with k elements in restricted positions, when the board of the restricted positions is a Ferrers board Bn, is derived as a backward finite difference of order n+1 of a modified polynomial of order n. Further, triangular recurrence relations for these numbers are provided. In particular, the cases of trapezoidal and Newcomb boards are examined. As applications, the numbers of permutations of a set, without or with repetitions, and a multiset, which have a given number of falls (or rises) are deduced.

Permutation tests for univariate or multivariate analysis of variance and regression

The most appropriate strategy to be used to create a permutation distribution for tests of individual terms in complex experimental designs is currently unclear. There are often many possibilities, including restricted permutation or permutation of some form of residuals. This paper provides a summary of recent empirical and theoretical results concerning available methods and gives recommendations for their use in univariate and multivariate applications. The focus of the paper is on complex designs in analysis of variance and multiple regression (i.e., linear models). The assumption of exchangeability required for a permutation test is assured by random allocation of treatments to units in experimental work. For observational data, exchangeability is tantamount to the assumption of independent and identically distributed errors under a null hypothesis. For partial regression, the method of permutation of residuals under a reduced model has been shown to provide the best test. For analysis of variance, one must first identify exchangeable units by considering expected mean squares. Then, one may generally produce either (i) an exact test by restricting permutations or (ii) an approximate test by permuting raw data or some form of residuals. The latter can provide a more powerful test in many situations.

A graph theoretic approach to feed-through pin assignment

We study in this paper the feed-through pin assignment problem for cell-based design that assigns global nets to feed-through cells or implicit feed-through ports. On a single row, for a restricted permutation of global nets, we develop a graph theoretic approach for optimally minimizing the density of the channel above (below) the row subject to the bound on the density of the channel below (above) the row. Based on this technique, an approach for iteratively improving style in which there is an over-the-cell feed-through layer.

Quantification of the Spatial Co-Occurrences of Ecological Boundaries

In this paper, we investigate spatial relationships between vegetation boundaries and environmental boundaries from a second-growth forest in southwestern Quebec, Canada. Four statistics that quantify the amount of direct spatial overlap and the mean minimum distance between boundaries are introduced and used to compute the degree of spatial co-occurrences between boundaries. The significance of these statistics is determined using randomized and restricted permutation tests. Boundaries based on tree species density are found to significantly overlap the locations of boundaries delineated by the environmental data at the study site. Significant overlap is also found using boundaries defined by tree presence-absence data and environmental variables. Vegetation boundaries based on tree species density and on tree presence-absence data are not, however, at the same locations. This suggests that for the study site the two types of vegetation boundaries (tree density and presence-absence) reflect different responses to underlying environmental processes. Vegetation boundaries determined using species diversity and species richness, although spatially related to the presence absence boundaries, did not overlap the environmental boundaries. Results of the two permutation tests (randomized and restricted) agree only when the spatial relationship between the two boundary types is strong. Overall, randomization is found to be a more conservative test for detecting boundary spatial relationships, rejecting the null hypothesis of no spatial relationship fewer times than the restricted permutation test.