Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube

The problem “Given a Boolean function [Formula: see text] of [Formula: see text] variables by its truth table vector. Find (if exists) a vector [Formula: see text] of maximal (or minimal) weight, such that [Formula: see text].” is considered here. It is closely related to the problem of computing the algebraic degree of Boolean functions which is an important cryptographic parameter. To solve this problem efficiently, we explore the orders of the vectors of the [Formula: see text]-dimensional Boolean cube [Formula: see text] according to their weights. The notion of “[Formula: see text]th layer” of [Formula: see text] is involved in the definition and examination of the “weight order” relation. It is compared with the known relation “precedes”. Several enumeration problems concerning these relations are solved and the relevant notes were added to three sequences in the on-line encyclopedia of integer sequences (OEIS). One special weight order is defined and examined in detail. In it, the lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called weight-lexicographic order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. The results obtained by them were used in creating two new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed — the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results from numerous tests confirm the efficiency of these algorithms. Other applications of the obtained algorithms are also discussed — when representing, generating and ranking other combinatorial objects.

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An On-Line Version of the Encyclopedia of Integer Sequences

The Electronic Journal of Combinatorics ◽

10.37236/1194 ◽

1994 ◽

Vol 1 (1) ◽

Cited By ~ 9

Author(s):

N. J. A. Sloane

Keyword(s):

Electronic Mail ◽

Lexicographic Order ◽

Academic Press ◽

Integer Sequences ◽

Annotated List ◽

New Material ◽

On Line

A Handbook of Integer Sequences was published by Academic Press in 1973, and contained an annotated list of 2372 sequences arranged in lexicographic order. Since then a great deal of new material has been added, and many improvements have been made to the original entries. The purpose of this note is to announce that the new version of the collection, The On-Line Encyclopedia of Integer Sequences can now be accessed by electronic mail.

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Lexicographic Unranking of Combinations Revisited

Algorithms ◽

10.3390/a14030097 ◽

2021 ◽

Vol 14 (3) ◽

pp. 97

Author(s):

Antoine Genitrini ◽

Martin Pépin

Keyword(s):

Time Complexity ◽

Complexity Analysis ◽

Lexicographic Order ◽

Familiar Word ◽

Integer Arithmetic ◽

Combinatorial Objects ◽

New Strategy ◽

High Level ◽

The Cost ◽

Multinomial Coefficients

In the context of combinatorial sampling, the so-called “unranking method” can be seen as a link between a total order over the objects and an effective way to construct an object of given rank. The most classical order used in this context is the lexicographic order, which corresponds to the familiar word ordering in the dictionary. In this article, we propose a comparative study of four algorithms dedicated to the lexicographic unranking of combinations, including three algorithms that were introduced decades ago. We start the paper with the introduction of our new algorithm using a new strategy of computations based on the classical factorial numeral system (or factoradics). Then, we present, in a high level, the three other algorithms. For each case, we analyze its time complexity on average, within a uniform framework, and describe its strengths and weaknesses. For about 20 years, such algorithms have been implemented using big integer arithmetic rather than bounded integer arithmetic which makes the cost of computing some coefficients higher than previously stated. We propose improvements for all implementations, which take this fact into account, and we give a detailed complexity analysis, which is validated by an experimental analysis. Finally, we show that, even if the algorithms are based on different strategies, all are doing very similar computations. Lastly, we extend our approach to the unranking of other classical combinatorial objects such as families counted by multinomial coefficients and k-permutations.

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Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

Algorithmica ◽

10.1007/s00453-019-00623-3 ◽

2019 ◽

Vol 82 (3) ◽

pp. 386-428 ◽

Cited By ~ 2

Author(s):

Andrei Asinowski ◽

Axel Bacher ◽

Cyril Banderier ◽

Bernhard Gittenberger

Keyword(s):

Kernel Method ◽

Lattice Paths ◽

Integer Sequences ◽

Unified Framework ◽

Analytic Combinatorics ◽

On Line ◽

Forbidden Patterns ◽

Motzkin Paths ◽

Self Avoiding Walks ◽

Pushdown Automata

Abstract In this article we develop a vectorial kernel method—a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges’s theorem.

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Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2723 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Mireille Bousquet-Mélou ◽

Anders Claesson ◽

Mark Dukes ◽

Sergey Kitaev

Keyword(s):

Generating Function ◽

Integer Sequences ◽

Combinatorial Objects ◽

New Class ◽

Chord Diagrams ◽

The Third ◽

Finite Series ◽

International Audience ◽

Ascent Sequences ◽

Pattern Avoiding Permutations

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

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Parameterised Enumeration for Modification Problems

Algorithms ◽

10.3390/a12090189 ◽

2019 ◽

Vol 12 (9) ◽

pp. 189 ◽

Cited By ~ 3

Author(s):

Nadia Creignou ◽

Raïda Ktari ◽

Arne Meier ◽

Julian-Steffen Müller ◽

Frédéric Olive ◽

...

Keyword(s):

Complexity Theory ◽

Lexicographic Order ◽

All Solutions ◽

Enumeration Problems ◽

Enumeration Algorithms ◽

Neighbourhood Structure ◽

Parameterised Complexity

Recently, Creignou et al. (Theory Comput. Syst. 2017), introduced the class Delay FPT into parameterised complexity theory in order to capture the notion of efficiently solvable parameterised enumeration problems. In this paper, we propose a framework for parameterised ordered enumeration and will show how to obtain enumeration algorithms running with an FPT delay in the context of general modification problems. We study these problems considering two different orders of solutions, namely, lexicographic order and order by size. Furthermore, we present two generic algorithmic strategies. The first one is based on the well-known principle of self-reducibility and is used in the context of lexicographic order. The second one shows that the existence of a neighbourhood structure among the solutions implies the existence of an algorithm running with FPT delay which outputs all solutions ordered non-decreasingly by their size.

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