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The permutation classes Av(1234,2341) and Av(1243,2314) [article]

David Bevan
2015 arXiv   pre-print
We investigate the structure of the two permutation classes defined by the sets of forbidden patterns 1234,2341 and 1243,2314.  ...  By considering how the Hasse graphs of permutations in these classes can be built from a sequence of rooted source graphs, we determine their algebraic generating functions.  ...  Acknowledgements Michael Albert's PermLab software [1] was of particular benefit in helping to visualise and explore the structure of permutations in the two permutation classes.  ... 
arXiv:1407.0570v3 fatcat:tmh7nwf7tzfmnouyj4diaoqldu

Counting Permutations that Avoid Many Patterns [article]

Yonah Biers-Ariel, Haripriya Chakraborty, John Chiarelli, Bryan Ek, Andrew Lohr, Jinyoung Park, Justin Semonsen, Richard Voepel, Mingjia Yang, Anthony Zaleski, Doron Zeilberger
2017 arXiv   pre-print
This paper presents a collection of experimental results regarding permutation pattern avoidance, focusing on cases where there are "many" patterns to be avoided.  ...  Table 1 . 1 Σ |Av n (Σ)| deg(P (n)) {1234, 1243, 1342, 4231} 1, 2, 6, 20, 64, 187, 492, 1170, 2543, 5116 6 {1234, 1243, 1432, 3412} 1, 2, 6, 20, 59, 148, 324, 638, 1157, 1966 5 {1234, 1243, 2341  ...  Example 2. 4 . 4 Let T = (2, 1, 3, 5, 4); then R 1,T = {1}, R 2,T = {12}, R 3,T = {123, 132, 213}, R 4,T = {1234, 1243, 1423, 2134, 2143, 2314}.  ... 
arXiv:1703.02415v1 fatcat:ov3oijkyozentblaedwerxuk3y

Descents and des-Wilf equivalence of permutations avoiding certain nonclassical patterns

Caden Bielawa, Robert Davis, Daniel Greeson, Qinhan Zhou
2019 Involve. A Journal of Mathematics  
In recent work of Dokos et al. the notion of Wilf equivalence was refined to reflect when avoidance of classical patterns preserves certain statistics.  ...  A frequent topic in the study of pattern avoidance is identifying when two sets of patterns Π, Π' are Wilf equivalent, that is, when |Av_n(Π)| = |Av_n(Π')| for all n.  ...  The permutation diagrammed in Figure 1 avoids the bivincular pattern 1234 3241 and contains 1234 3241 .  ... 
doi:10.2140/involve.2019.12.549 fatcat:7oebsesqtfdmblhg4yo2gxxype

Vincular pattern avoidance on cyclic permutations

Rupert Li, Massachusetts Institute of Technology
2021 Enumerative Combinatorics and Applications  
Further, several of the avoidance classes corresponding to a single vincular pattern of length 4 are enumerated by the Catalan numbers.  ...  Pattern avoidance for permutations has been extensively studied and has been generalized to vincular patterns, where certain elements can be required to be adjacent.  ...  Joe Gallian for his editing feedback and operation of the Duluth REU program.  ... 
doi:10.54550/eca2022v2s4pp3 fatcat:d4mpzj6tavhanbc6isdoglu6au

Vincular Pattern Avoidance on Cyclic Permutations [article]

Rupert Li
2022 arXiv   pre-print
Further, several of the avoidance classes corresponding to a single vincular pattern of length 4 are enumerated by the Catalan numbers.  ...  Pattern avoidance for permutations has been extensively studied, and has been generalized to vincular patterns, where certain elements can be required to be adjacent.  ...  Joe Gallian for his editing feedback and operation of the Duluth REU program.  ... 
arXiv:2107.12353v2 fatcat:566evcn7kzfhfbi4wrvpslkvdu

Algorithms for discovering and proving theorems about permutation patterns [article]

Hjalti Magnusson, Henning Ulfarsson
2012 arXiv   pre-print
It automatically conjectures the statements of known theorems such as the descriptions of stack-sortable (Knuth 1975) and West-2-stack-sortable permutations (West 1990), smooth (Lakshmibai and Sandhya  ...  1990) and forest-like permutations (Bousquet-Melou and Butler 2007), and simsun permutations (Branden and Claesson 2011).  ...  He would finally like to thank Einar Steingrímsson for many helpful discussions on permutation patterns. The name of the algorithm is derived from the names of these three people.  ... 
arXiv:1211.7110v1 fatcat:alj5ke4g4jeoxp4fk5ezoz3rzu

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

David Callan, Toufik Mansour
2018 Pure Mathematics and Applications  
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes.  ...  (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima.  ...  Now, suppose i(n − 1)π ′′ avoids T , so if π = j(n − 1)iπ ′′ contains 1234 or 1243, then i(n − 1)π ′′ contains 1234 or 1243, which implies that π avoids 1234 and 1243.  ... 
doi:10.1515/puma-2015-0026 fatcat:qhjwk2ujp5glneepcyvmh6xo24

Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns [article]

David Callan, Toufik Mansour
2017 arXiv   pre-print
Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated.  ...  We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function  ...  Then π (2) > i 1 (2314) and π (3) < i 2 (1243).  ... 
arXiv:1708.00832v1 fatcat:vvghd5zmj5dytjhi6jmiclrs6a

On the growth of permutation classes [article]

David Bevan
2015 arXiv   pre-print
We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class.  ...  We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations.  ...  AVOIDING 1324 AND 2314 . AVOIDING 1324 AND 2314 . PLANE PERMUTATIONS Also known as Lehmer codes. . AVOIDING 1234 AND 2341 . AVOIDING 1234 AND 2341 .  ... 
arXiv:1506.06688v1 fatcat:jgznt4crvrdt5dcjqiaelwdo5m

Describing West-3-stack-sortable permutations with permutation patterns [article]

Henning Úlfarsson
2012 arXiv   pre-print
The method requires the use of mesh patterns introduced by Br\"and\'en and Claesson (2011), as well as a new type of generalized pattern we call a decorated pattern.  ...  We describe a new method for finding patterns in permutations that produce a given pattern after the permutation has been passed once through a stack.  ...  The author would also like to thank Christian Krattenthaler and an anonymous referee for detailed comments. The author is supported by grant no. 090038013 from the Icelandic Research Fund.  ... 
arXiv:1110.1219v3 fatcat:mgr2gwkstvblnd6kttc3dhdfau

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns [article]

David Callan, Toufik Mansour
2017 arXiv   pre-print
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes.  ...  (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima.  ...  Acknowledgement We thank Richard Mather for pointing out some inaccuracies in the previous version.  ... 
arXiv:1705.00933v3 fatcat:sfdjeay5hfgefipgr4lulq2tce

Automatic discovery of structural rules of permutation classes [article]

Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson
2017 arXiv   pre-print
The algorithm is successful on different inputs than other algorithms and can succeed with any polynomial permutation class.  ...  permutation class.  ...  them, e.g. 213 and 1234 will generate the permutation 21384567.  ... 
arXiv:1705.04109v1 fatcat:sjykf6tq4zfuvggdvodsu7no4a

Definite Determinantal Representations via Orthostochastic Matrices [article]

Papri Dey
2019 arXiv   pre-print
We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where  ...  In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some  ...  P 4312 , P 3421 }, B 4 := {P 3214 , P 1432 , P 2341 , P 4123 }, B 5 := {P 4231 , P 1324 , P 3142 , P 2413 }, B 6 := {P 1234 , P 2143 , P 3412 , P 4321 , } B 7 := {P 1234 , P 2134 , P 1243 , P 2143 }, B  ... 
arXiv:1708.09559v3 fatcat:n2cootfgxffldi4drcv6m4q5re

Laplacian spectrum for the nilpotent Kac–Moody Lie algebras

Dmitry Fuchs, Constance Wilmarth
2010 Pacific Journal of Mathematics  
Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.  ...  We prove that a maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric whose Laplace operator in the chain complex is scalar on each component of a given degree  ...  Similarly for n(4) (we write σ (i jkl) for the permutation (i, j, k, l)): The cohomology classes of the corresponding monomial cochains form a basis in the cohomology: γ (1234) = 1 ∈ H 0 (n(4)), γ (2134  ... 
doi:10.2140/pjm.2010.247.323 fatcat:i3m5ol3vj5eijiy3dtk5jxgeia

Invariants of multidimensional time series based on their iterated-integral signature [article]

Joscha Diehl, Jeremy Reizenstein
2018 arXiv   pre-print
The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen's iterated-integral signature.  ...  We introduce a novel class of features for multidimensional time series, that are invariant with respect to transformations of the ambient space.  ...  − 1243 − 4123 − 2413 + 1234 + 3124 + 4213 + 1342 + 2431 − 2134 + 2143 + 3241 + 2314 + 4321 − 4312 − 3214 Denote by S d the group of permutations of [d] := {1, .., d}.  ... 
arXiv:1801.06104v2 fatcat:a3uxoe3oevcnbm7vbkkpvnfieu
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