scholarly journals Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns

10.37236/1687 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Yu-Ping Deng ◽  
Laura L. M. Yang
Keyword(s):  
Recent Result ◽  
Motzkin Path ◽  
Reduced Decompositions ◽  
Inversion Number ◽  
Forbidden Patterns ◽  
Motzkin Paths

We obtain a characterization of $(321, 3\bar{1}42)$-avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of $(321,3\bar{1}42)$-avoiding permutations of length $n$ equals the $n$-th Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo, Pergola, Pinzani and Guibert. Similarly, we obtain a characterization of $(231,4\bar{1}32)$-avoiding permutations. For these two classes, we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover, we find a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path.

scholarly journals ON THE NUMBER OF ABELIAN BORDERED WORDS (WITH AN EXAMPLE OF AUTOMATIC THEOREM-PROVING)

2014 ◽  
Vol 25 (08) ◽  
pp. 1097-1110 ◽  
Author(s):  
DANIEL GOČ ◽  
NARAD RAMPERSAD ◽  
MICHEL RIGO ◽  
PAVEL SALIMOV
Keyword(s):  
Theorem Proving ◽  
Triangular Lattice ◽  
Automatic Theorem Proving ◽  
Combinatorial Objects ◽  
Letter Alphabet ◽  
Automatic Sequences ◽  
Motzkin Paths ◽  
Automatic Theorem

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in ℤ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue–Morse word using some kind of automatic theorem-proving provided by a logical characterization of the k-automatic sequences.

scholarly journals Orders Induced by Segments in Floorplans and $(2 - 14 - 3, 3 - 41 - 2)$-Avoiding Permutations

10.37236/2607 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Andrei Asinowski ◽  
Gill Barequet ◽  
Mireille Bousquet-Mélou ◽  
Toufik Mansour ◽  
Ron Y. Pinter
Keyword(s):  
The Sixties ◽  
Neighborhood Relations ◽  
Forbidden Patterns ◽  
Special Case

A floorplan is a tiling of a rectangle by rectangles. There are natural ways to order the elements - rectangles and segments - of a floorplan. Ackerman, Barequet and Pinter studied a pair of orders induced by neighborhood relations between rectangles, and obtained a natural bijection between these pairs and $(2 - 41 - 3, 3 - 14 - 2)$-avoiding permutations, also known as (reduced) Baxter permutations.In the present paper, we first perform a similar study for a pair of orders induced by neighborhood relations between segments of a floorplan. We obtain a natural bijection between these pairs and another family of permutations, namely $(2 - 14 - 3, 3 - 41 - 2)$-avoiding permutations.Then, we investigate relations between the two kinds of pairs of orders - and, correspondingly, between $(2 - 41 - 3, 3 - 14 - 2)$- and $(2 - 14 - 3, 3 - 41 - 2)$-avoiding permutations. In particular, we prove that the superposition of both permutations gives a complete Baxter permutation (originally called $w$-admissible by Baxter and Joichi in the sixties). In other words, $(2 - 14 - 3, 3 - 41 - 2)$-avoiding permutations are the hidden part of complete Baxter permutations. We enumerate these permutations. To our knowledge, the characterization of these permutations in terms of forbidden patterns and their enumeration are both new results.Finally, we also study the special case of the so-called guillotine floorplans.

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

Algorithmica ◽  
2019 ◽  
Vol 82 (3) ◽  
pp. 386-428 ◽  
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.

scholarly journals Further Results on Recursive Evaluation of Compound Distributions

1981 ◽  
Vol 12 (1) ◽  
pp. 27-39 ◽  
Author(s):  
Bjørn Sundt ◽  
William S. Jewell
Keyword(s):  
Recent Result ◽  
Recursive Algorithm ◽  
Compound Distributions ◽  
Compound Distribution ◽  
Aggregate Claims ◽  
Recursive Evaluation

A recent result by Panjer provides a recursive algorithm for the compound distribution of aggregate claims when the counting law belongs to a special recursive family. In the present paper we first give a characterization of this recursive family, then describe some generalizations of Panjer's result.

scholarly journals Minimal and maximal plateau lengths in Motzkin paths

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Gumbel Distribution ◽  
Minimal Length ◽  
Analytic Method ◽  
Horizontal Segment ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Substitution Process ◽  
International Audience ◽  
Motzkin Paths

International audience The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters <i> maximal plateau length, horizontal segment </i>and <i>maximal horizontal segment </i>as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

A Characterization of Line Spaces

1981 ◽  
Vol 24 (3) ◽  
pp. 273-277 ◽  
Author(s):  
J. H. M. Whitfield ◽  
S. Yong
Keyword(s):  
Recent Result ◽  
Linearization Problem

AbstractThe line spaces of J. Cantwell are characterized among the axiomatic convexity spaces defined by Kay and Womble. This characterization is coupled with a recent result of Doignon to give an intrinsic solution of the linearization problem.

scholarly journals Applications of intuitionistic logic in Answer Set Programming

2004 ◽  
Vol 4 (3) ◽  
pp. 325-354 ◽  
Author(s):  
MAURICIO OSORIO ◽  
JUAN A. NAVARRO ◽  
JOSÉ ARRAZOLA
Keyword(s):  
Recent Result ◽  
Intuitionistic Logic ◽  
Answer Set Programming ◽  
Minimal Models ◽  
Logic Programs ◽  
Intermediate Logics ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.

scholarly journals CLASSIFICATION THEORY FOR ACCESSIBLE CATEGORIES

2016 ◽  
Vol 81 (1) ◽  
pp. 151-165 ◽  
Author(s):  
M. LIEBERMAN ◽  
J. ROSICKÝ
Keyword(s):  
Recent Result ◽  
Large Cardinal ◽  
Classification Theory ◽  
Abstract Elementary Classes ◽  
Robust Version ◽  
Accessible Categories ◽  
Elementary Classes ◽  
Added Benefit ◽  
Precise Role

AbstractWe show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht–Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

scholarly journals Permutations realized by shifts

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde
Keyword(s):  
Relative Order ◽  
Letter Alphabet ◽  
Infinite Word ◽  
Set Of Permutations ◽  
International Audience ◽  
Forbidden Patterns ◽  
Pattern Containment

International audience A permutation $\pi$ is realized by the shift on $N$ symbols if there is an infinite word on an $N$-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J.M. Amigó, S. Elizalde and M. Kennel, $\textit{J. Combin. Theory Ser. A}$ 115 (2008), 485―504] that the shortest forbidden patterns of the shift on $N$ symbols have length $N+2$. In this paper we give a characterization of the set of permutations that are realized by the shift on $N$ symbols, and we enumerate them with respect to their length. Une permutation $\pi$ est réalisée par le $\textit{shift}$ avec $N$ symboles s'il y a un mot infini sur un alphabet de $N$ lettres dont les déplacements successifs d'une position à gauche sont lexicographiquement dans le même ordre relatif que $\pi$. Les permutations qui ne sont pas réalisées s'appellent des motifs interdits. On sait [J.M. Amigó, S. Elizalde and M. Kennel, $\textit{J. Combin. Theory Ser. A}$ 115 (2008), 485―504] que les motifs interdits les plus courts du $\textit{shift}$ avec $N$ symboles ont longueur $N+2$. Dans cet article on donne une caractérisation des permutations réalisées par le $\textit{shift}$ avec $N$ symboles, et on les dénombre selon leur longueur.

scholarly journals Reduced Decompositions of Matchings

10.37236/594 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Lun Lv ◽  
Sabrina X. M. Pang
Keyword(s):  
Reduced Decompositions

We give a characterization of matchings in terms of the canonical reduced decompositions. As an application, the canonical reduced decompositions of $12312$-avoiding matchings are obtained. Based on such decompositions, we find a bijection between $12312$-avoiding matchings and ternary paths.

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