15 research outputs found
Juxtaposing Catalan Permutation Classes with Monotone Ones
This paper enumerates all juxtaposition classes of the form "Av(abc) next to Av(xy)", where abc is a permutation of length three and xy is a permutation of length two. We use Dyck paths decorated by sequences of points to represent elements from such a juxtaposition class. Context free grammars are then used to enumerate these decorated Dyck paths
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Packing and Counting Permutations
A permutation class is a set of permutations closed under taking subpermutations. We study two aspects of permutation classes: enumeration and packing.
Our work on enumeration consists of two campaigns. First, we enumerate all juxtaposition classes of the form “Av(abc) next to Av(xy)”, where abc and xy are permutations of lengths three and two, respectively. We represent elements from such a juxtaposition class by Dyck paths decorated with sequences of points. Context-free grammars are then used to enumerate these decorated Dyck paths. Second, we classify as algebraic the generating functions of 1×m permutation grid classes where one cell is context-free and the remaining cells are monotone. We rely on properties of combinatorial specifications of context-free classes and use operators to express juxtapositions. Repeated application of operators resolves cases for m > 2. We provide examples to re-prove known results and give new ones. Our methods are algorithmic and could be implemented on a PC.
Our work on packing consolidates current knowledge about packing densities of 4-point permutations. We also improve the lower bounds for the packing densities of 1324 and 1342 and provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within 10-4 of the true packing densities. Together with the known bounds, we have a fairly complete picture of 4-point packing densities. Additionally, we obtain several bounds (lower and upper) for permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007. We also present Permpack — a flag algebra package for permutations
Combinatorial specifications for juxtapositions of permutation classes
We show that, given a suitable combinatorial specification for a permutation class C, one can obtain a specification for the juxtaposition (on either side) of C with Av(21) or Av(12), and that if the enumeration for C is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any "skinny" k×1 grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of the enumeration of the non-monotone cell
Bounded affine permutations I. Pattern avoidance and enumeration
We introduce a new boundedness condition for affine permutations, motivated
by the fruitful concept of periodic boundary conditions in statistical physics.
We study pattern avoidance in bounded affine permutations. In particular, we
show that if is one of the finite increasing oscillations, then every
-avoiding affine permutation satisfies the boundedness condition. We also
explore the enumeration of pattern-avoiding affine permutations that can be
decomposed into blocks, using analytic methods to relate their exact and
asymptotic enumeration to that of the underlying ordinary permutations.
Finally, we perform exact and asymptotic enumeration of the set of all bounded
affine permutations of size . A companion paper will focus on avoidance of
monotone decreasing patterns in bounded affine permutations.Comment: 35 page
Finding structure in permutation sets
New automatic methods for enumerating permutation classes are introduced. The first is Struct, which is an algorithm that conjectures a structural description using rules similar to generalized grid classes. These conjectured structural descriptions can be easily enumerated and are easily verified by a human to be correct.
We then introduce the CombSpecSearcher algorithm, a general framework for searching for combinatorial specifications. To use this, one must write strategies that explain how combinatorial classes are related. We introduce strategies for permutation classes that are used by the CombSpecSearcher algorithm.
We provide an algorithm for finding the insertion encoding of regular insertion encodable permutation classes. Our approach requires less generation of permutations and as such is much faster than previous implementations. The algorithm relies on tilings; a new object introduced that can be used to encode geometric proof ideas for permutation patterns efficiently.
After developing the theory of tilings briefly, we encode more geometric proof ideas that allow for placing points, each such leading to a new algorithm. All of these algorithms search for combinatorial specifications using the CombSpecSearcher which can then be enumerated. The algorithms introduced that use CombSpecSearcher and tilings we collectively call TileScope.
We introduce the notion of an elementary permutation class. We show that our definition is equivalent to the permutation class being a disjoint union of generalized peg permutations, and as such all polynomial permutation classes are elementary. We show that such a description for a permutation class can be extended to permutation classes where the basis has an extra pattern.
The methods introduced in this thesis can enumerate all permutation classes with six or more length four patterns. There are only 77 bases consisting of only length four patterns which are not enumerated by our methods.Nýjar sjálfvirkar aðferðir til að telja umraðanaflokka eru kynntar. Sú fyrsta erStruct, sem erreiknirit til að búa til tilgátur um uppbyggingu flokka með alhæfðum grindargerðum. Þessartilgátur er auðvelt að nota til talninga og staðfesta af manneskju.Síðan kynnum viðCombSpecSearcher, almenna umgjörð til að leita að fléttufræðilegumforskriftum. Til að nota umgjörðina þarf að skrifa kænskur sem útskýra hvernig fléttu-fræðilegir flokkar tengjast. Við kynnum kænskur fyrir umraðanaflokka og notum þær íumgjörðinni.Við gefum reiknirit til að finnainnsetningarumrituninafyrir umraðanaflokka sem hafa reglu-lega innsetningarumritun. Aðferð okkar þarfnast minni framleiðslu umraðana og er þvíhraðvirkari en fyrri útfærslur. Reikniritið byggir áflísum; nýs hlutar sem hægt er að um-rita rúmfræðilegar sönnunaraðferðir með. Eftir að hafa þróað fræði flísa þá umritum viðsönnunaraðferðir til þess að staðsetja punkta og fáum þannig ný reiknirit. Öll þessi reikniritnýtaCombSpecSearcher-umgjörðina og leiða til talninga. Við gefum þessum reikniritumyfirheitiðTilescope.Við kynnumgrundvallarumraðanaflokkaog sýnum að það eru þeir umraðanaflokkar semeru ósamsniða sammengi alhæfðra pinnaumraðana. Þar af leiðandi eru allir margliðu-umraðanaflokkar grundvallarumraðanaflokkar. Við sýnum að viðbót á mynstri við grunngrundvallarumraðanaflokks gefur annan grundvallarumraðanaflokk.Aðferðirnar í þessari ritgerð geta talið alla umraðanflokka með sex eða fleiri mynstur af lengdfjórum. Það eru einungis77grunnar með mynstur af lengd fjórum sem aðferðir okkar getaekki talið.The research in this thesis was partially supported by grant 141761-051 from the Icelandic Research Fund
Advanced rank/select data structures: succinctness, bounds and applications.
The thesis explores new theoretical results and applications of rank and select data structures. Given a string, select(c, i) gives the position of the ith occurrence of character c in the string, while rank(c, p) counts the number of instances of character c on the left of position p.
Succinct rank/select data structures are space-efficient versions of standard ones, designed to keep data compressed and at the same time answer to queries rapidly. They are at the basis of more involved compressed and succinct data structures which in turn are motivated by the nowadays need to analyze and operate on massive data sets quickly, where space efficiency is crucial. The thesis builds up on the state of the art left by years of study and produces results on multiple fronts.
Analyzing binary succinct data structures and their link with predecessor data structures, we integrate data structures for the latter problem in the former. The result is a data structure which outperforms the one of Patrascu 08 in a range of cases which were not studied before, namely when the lower bound for predecessor do not apply and constant-time rank is not feasible.
Further, we propose the first lower bound for succinct data structures on generic strings, achieving a linear trade-off between time for rank/select execution and additional space (w.r.t. to the plain data) needed by the data structure. The proposal addresses systematic data structures, namely those that only access the underlying string through ADT calls and do not encode it directly.
Also, we propose a matching upper bound that proves the tightness of our lower bound.
Finally, we apply rank/select data structures to the substring counting problem, where we seek to preprocess a text and generate a summary data structure which is stored in lieu of the text and answers to substring counting queries with additive error. The results include a theory-proven optimal data structure with generic additive error and a data structure that errs only on infrequent patterns with significative practical space gains
Quanta of Maths
The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics
Quanta of Maths
The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics
Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education
International audienceThis volume contains the Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (ERME), which took place 9-13 February 2011, at Rzeszñw in Poland