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# PatternClass

## Permutation Pattern Classes

Version 2.0

August 2015

Michael Albert
Email: malbert at cs.otago.ac.nz

Steve Linton
Email: sl4 at st-andrews.ac.uk

Ruth Hoffmann
Email: rh347 at icloud.com

### Abstract

The `PatternClass` package is build on the idea of token passing networks building permutation pattern classes. Those classes are best determined by their basis. Both sets can be encoded by rank encoding their permutations. Each, the class and its basis, in their encoded form build a rational language [AAR03]. Rational languages can be easily computed by using automata, which also can be build directly from the token passing networks [ALTd.]. Both ways will build the same language, i.e. the same automaton.

### Copyright

© 2011 by Michael Albert, Steve Linton and Ruth Hoffmann

### Contents

1 Introduction
2 Token Passing Networks
2.1 Specific TPN
2.1-1 Parstacks
2.1-2 Seqstacks
2.1-3 BufferAndStack
3 Permutation Encoding
3.1 Encoding and Decoding
3.1-1 RankEncoding
3.1-2 RankDecoding
3.1-3 SequencesToRatExp
4 From Networks to Automata
4.1 Functions
4.1-1 GraphToAut
4.1-2 ConstrainedGraphToAut
5 From Automata to Networks
5.1 Functions
5.1-1 IsStarClosed
5.1-2 Is2StarReplaceable
5.1-3 IsStratified
5.1-4 IsPossibleGraphAut
6 Pattern Classes
6.1 Transducers
6.1-1 Transducer
6.1-2 DeletionTransducer
6.1-3 TransposedTransducer
6.1-4 InvolvementTransducer
6.1-5 CombineAutTransducer
6.2 From Class to Basis and vice versa
6.2-1 BasisAutomaton
6.2-2 ClassAutomaton
6.2-3 BoundedClassAutomaton
6.2-4 ClassAutFromBaseEncoding
6.2-5 ClassAutFromBase
6.2-6 ExpandAlphabet
6.3 Direct Sum of Regular Classes
6.3-1 ClassDirectSum
6.4 Statistical Inspections
6.4-1 Spectrum
6.4-2 NumberAcceptedWords
6.4-3 AutStateTransitionMatrix
6.4-4 AcceptedWords
6.4-5 AcceptedWordsR
7 Some Permutation Essentials
7.1 Complement
7.1-1 PermComplement
7.2 Rank Encoding
7.2-1 IsRankEncoding
8 Properties of Permutations
8.1 Intervals in Permutations
8.1-1 IsInterval
8.2 Simplicity
8.2-1 IsSimplePerm
8.3 Point Deletion in Simple Permutations
8.3-1 OnePointDelete
8.3-2 TwoPointDelete
8.3-3 PointDeletion
8.4 Block-Decomposition
8.4-1 Inflation
8.4-2 BlockDecomposition
8.5 Plus-Decomposability
8.5-1 IsPlusDecomposable
8.6 Minus-Decomposability
8.6-1 IsMinusDecomposable
8.7 Sums of Permutations
8.7-1 PermDirectSum
8.7-2 PermSkewSum
9 Regular Languages of Sets of Permutations
9.1 Inversions in Permutations
9.1-1 InversionAut
9.1-2 InversionAutOfClass
9.2 Plus- and Minus-(In)Decomposablilty
9.2-1 PlusDecomposableAut
9.2-2 PlusIndecomposableAut
9.2-3 MinusDecomposableAut
9.2-4 MinusIndecomposableAut
9.3 Language of all non-simple permutations
9.3-1 LengthBoundAut
9.3-2 ShiftAut
9.3-3 NextGap
9.3-4 GapAut
9.3-5 SumAut
9.3-6 GapSumAut
9.3-7 NonSimpleAut
9.4 Simplicity
9.4-1 SimplePermAut
9.5 Exceptionality
9.5-1 IsExceptionalPerm
9.5-2 ExceptionalBoundedAutomaton
10 Miscellaneous functions
10.1 Automaton Manipulation
10.1-1 LoopFreeAut
10.1-2 LoopVertexFreeAut
Index
References
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