Multiset Permutations Research Articles

Generalized Rook Polynomials

Generalizing the notion of placing rooks on a Ferrers board leads to a new class of combinatorial models and a new class of rook polynomials. Connections are established with absolute Stirling numbers and permutations, Bessel polynomials, matchings, multiset permutations, hypergeometric functions, Abel polynomials and forests, and polynomial sequences of binomial type. Factorization and reciprocity theorems are proved and a q-analogue is given.

Multiset Permutations and Loopless Generation of Ordered Trees with Specified Degree Sequence

An ordered tree with specified degree sequence and n internal nodes has ai nodes of degree i, where a0=1+∑i=1(i−1)ai and n=∑i=0ai. This paper presents the first loopless algorithm for generating all ordered trees with specified degree sequence. It uses a new version of the algorithm for generating multiset permutations. When ak=N, a0=(k−1)N+1, and all other ai's are 0, all N node k-ary trees are generated.

Permutation (d,k) codes: efficient enumerative coding and phrase length distribution shaping

We introduce a new enumerative encoding method for (d,k) codes. The encoding algorithm, which is based on enumeration of multiset permutations, is conceptually simpler and computationally less expensive than other algorithms proposed thus far. We also describe a new application of enumerative encoding methods for phrase length distribution shaping of run-length-limited (RLL) sequences. We demonstrate that by reducing the probability of occurrence of long phrases in maxentropic RLL sequences, the frequency of patterns that account for most of the errors in magnetic recording systems can be decreased.

Generating Multiset Permutations in Constant Time

An algorithm is presented that generates multiset permutations taking constant time between each permutation.

Application of inversions to lossless image compression

Linear prediction schemes, such as that of the Joint Photographic Experts Group (JPEG), are simple and normally produces a residual sequence with lower zero-order entropy. Occasionally the entropy of the prediction error becomes greater than that of the original image. Such situations frequently occur when the image data have discrete gray levels located within certain intervals. To alleviate this problem, various authors have suggested different preprocessing methods. However, the techniques reported require two passes. We extend the definition of Lehmer-type inversions (Lehmer 1960 and 1964) from permutations to multiset permutations and present a one-pass algorithm based on inversions of a multiset permutation. We obtain comparable results when we apply JPEG and even better results when we apply some other linear prediction schemes on a preprocessed image, which is treated as multiset permutation.

Correction to Algorithm AS 179: Enumeration of All Permutations of Multi-sets with Fixed Repetition Numbers

Correction to Algorithm AS 179: Enumeration of All Permutations of Multi-sets with Fixed Repetition Numbers

Inverse halftoning using binary permutation filters

The problem of reconstructing a continuous-tone image given its ordered dithered halftone or its error-diffused halftone image is considered. We develop a modular class of nonlinear filters that can reconstruct the continuous-tone information preserving image details and edges that provide important visual cues. The proposed nonlinear reconstruction algorithms, denoted as binary permutation filters, are based on the space and rank orderings of the halftone samples provided by the multiset permutation of the "on" pixels in a halftone observation window. For a given window size, we obtain a wide range of filters by varying the amount of space-rank ordering information utilized in the estimate. For image reconstructions from ordered dithered halftones, we develop periodically space-varying filters that can account for the periodical nature of the underlying screening process. A class of suboptimal but simpler space-invariant reconstruction filters are also proposed and tested. Constrained LMS type algorithms are employed for the design of reconstruction filters that minimize the reconstruction mean squared error. We present simulations showing that binary permutation filters are modular, robust to image source characteristics, and that they produce high visual quality image reconstruction.

On a Theorem of Janson

Previous article Next article On a Theorem of JansonV. G. MikhailovV. G. Mikhailovhttps://doi.org/10.1137/1136018PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Svante Janson, Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs, Ann. Probab., 16 (1988), 305–312 89a:60062 0639.60029 CrossrefGoogle Scholar[2] M. V. Petrovskaya and , A. M. Leontovich, The central limit theorem for a sequence of random variables with a slowly growing number of dependencies, Theory Probab. Appl., 27 (1982), 815–825 10.1137/1127089 0559.60069 LinkGoogle Scholar[3] A. M. Leontovich, On a condition for slowly growing number of dependencies, Theory Probab. Appl., 30 (1985), 196–200 10.1137/1130026 0657.60015 LinkGoogle Scholar[4] V. G. Mikhailov, On the asymptotic normality of U-statistics with non-negative kernels, Proc. of the Steklov Math. Inst., 1988, 105–113, issue 4 Google Scholar[5] V. P. Leonov and , A. N. Shiryaev, On a method of semi-invariants, Theor. Probability Appl., 4 (1959), 319–329 23:A673 0087.33701 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Об условиях асимптотической нормальности числа повторений в стационарной случайной последовательности28 August 2021 | Дискретная математика, Vol. 33, No. 3 Cross Ref Central limit theorems for patterns in multiset permutations and set partitionsThe Annals of Applied Probability, Vol. 30, No. 1 Cross Ref Weighted dependency graphsElectronic Journal of Probability, Vol. 23, No. none Cross Ref Dependency graphs and mod-Gaussian convergence7 December 2016 Cross Ref Asymptotic normality of the number of values of m-dependent random variables which occur a given number of timesDiscrete Mathematics and Applications, Vol. 21, No. 1 Cross Ref Asymptotic normality of the number of absent noncontinuous chains of outcomes of independent trialsDiscrete Mathematics and Applications, Vol. 19, No. 3 Cross Ref A Refinement of the Central Limit Theorem for Sums of Independent Random IndicatorsA. Yu. Volkova12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 4AbstractPDF (442 KB) Volume 36, Issue 1| 1992Theory of Probability & Its Applications History Submitted:30 January 1989Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1136018Article page range:pp. 173-176ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Q-Hook length formulas for forests

We present two q-analogues of a hook length formula of Knuth for the number of linear extensions of a partially ordered set whose Hasse diagram is a rooted forest. These q-analogues give formulas for the inversion index and the major index generating functions over permutations which correspond to linear extensions of a labeled forest. They generalize and unify several other q-formulas appearing in the literature. For linear forests all of these formulas reduce to MacMahon's classical formula for “ q-counting” multiset permutations according to the major index and inversion index. We also extend MacMahon's formula in another direction by q-counting all labelings of a fixed forest according to two very natural statistics on labeled forests which generalize the major index and inversion index on permutations.

Solution of some multi-dimensional lattice path parity difference recurrence relations

Many elementary combinatorial objects can be modeled as restricted multiset permutations. Standard examples include (unrestricted) permutations of a multiset and ordered trees with prescribed degree sequence. In developing algorithms to generate all members of a subset S of multiset permutations in such a way that successive permutations differ only by the interchange of adjacent distinct elements, we are led to the adjacent interchange graph G( S) of S. The vertices of G( S) are the elements of S and two vertices are connected by an edge if one can be obtained from the other by an adjacent interchange. This graph is bipartite. no adjacent interchange algorithm is possible if the number of vertices in the two bipartitions differs by more than one. In this paper we determine a recurrence relation that describes the difference of the number of vertices in the two bipartitions. This recurrence relation is solved for the examples mentioned above.

Algorithm AS 179: Enumeration of All Permutations of Multi-Sets with Fixed Repetition Numbers

Algorithm AS 179: Enumeration of All Permutations of Multi-Sets with Fixed Repetition Numbers

A lattice walk approach to the “inv” and “maj” q-counting of multiset permutations

A natural interpretation of “maj” and “inv” q-counting of multiset permutations in terms of walks on a lattice with multilane highways is presented. This is applied to give a short combinatorial proof of two theorems of MacMahon and to rederive a recent result of Gessel.

Binomial posets, Möbius inversion, and permutation enumeration

A unified method is presented for enumerating permutations of sets and multisets with various conditions on their descents, inversions, etc. We first prove several formal identities involving Möbius functions associated with binomial posets. We then show that for certain binomial posets these Möbius functions are related to problems in permutation enumeration. Thus, for instance, we can explain “why” the exponential generating function for alternating permutations has the simple form ( 1 + sin x) ( cos x) . We can also clarify the reason for the ubiquitous appearance of e x in connection with permutations of sets, and of ξ( s) in connection with permutations of multisets.