Automatic discovery of structural rules of permutation classes

Bean, Christian; Guðmundsson, Bjarki Ágúst; Ulfarsson, Henning

doi:10.1090/mcom/3386

Cited by 4 publications

(4 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…permutation in Av 3 (Π), namely 132. The same structure for the decomposition of the permutations in Av(Π) was recently found with a computer by Bean et al who used a particular algorithm called the Struct algorithm [5]. As we saw, rewriting the problem in terms of distant patterns helped us to prove the result directly and to give an interpretation of the already discovered decomposition.…”

Section: Open Problems and Future Worksupporting

confidence: 68%

On permutation patterns with constrained gap sizes

Dimitrov¹

2020

Preprint

View full text Add to dashboard Cite

We consider avoidance of permutation patterns with designated gap sizes between pairs of consecutive letters. We call the patterns having such constraints distant patterns (DPs) and we show their relation to other pattern notions investigated in the past. New results on DPs with 2 and 3 letters are obtained including a generating function found using the block-decomposition method in a non-trivial way. Furthermore, we prove two conjectures of Kuszmaul using a DP interpretation and we give that perspective to four of the other conjectures listed there. DPs with tight gap constraints are also considered in order to deduce a somewhat surprising relation between the sets of permutations avoiding the classical patterns 123 and 132. In addition, interesting Stanley-Wilf analogues for DPs are discussed, as well as some open questions.

show abstract

Section: Open Problems and Future Worksupporting

confidence: 68%

On permutation patterns with constrained gap sizes

Dimitrov¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…As the figure shows, several of these methods, e.g., Zeilberger's enumeration schemes, Vatter's implementation of the regular insertion encoding, and the substitution decomposition applied to permutation classes with finitely many simples, are subsumed by Combinatorial Exploration. We believe it may be possible that Polynomial classes [90] Finitely labeled generating tree [136] Templates [31] Struct-cover verified [23] Regular insertion encoding [135] Zeilberger's finite enum. scheme [139] Scanning elements algorithm [79] Finitely many simple permutations [20] Vatter's finite enum.…”

Section: Enumeration Of Permutation Classesmentioning

confidence: 99%

Combinatorial Exploration: An algorithmic framework for enumeration

Albert¹,

Bean²,

Claesson³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python implementation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications.We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones. These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com. Finally, we give three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions.

show abstract

“…Moreover, there is a connection to the so-called insertion-encodable classes which appear often in the area of permutation classes enumeration, see e.g. [5,7,26].…”

Section: Theorem 41 Let M Be a Gridding Matrix Such That Every Entry ...mentioning

confidence: 99%

A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes

Jelínek,

Opler,

Pekárek

2020

Preprint

View full text Add to dashboard Cite

Permutation Pattern Matching (PPM) is the problem of deciding for a given pair of permutations π and τ whether the pattern π is contained in the text τ . Bose, Buss and Lubiw showed that PPM is NP-complete. In view of this result, it is natural to ask how the situation changes when we restrict the pattern π to a fixed permutation class C; this is known as the C-Pattern PPM problem. There have been several results in this direction, namely the work of Jelínek and Kynčl who completely resolved the hardness of C-Pattern PPM when C is taken to be the class of σ-avoiding permutations for some σ.Grid classes are special kind of permutation classes, consisting of permutations admitting a grid-like decomposition into simpler building blocks. Of particular interest are the so-called monotone grid classes, in which each building block is a monotone sequence. Recently, it has been discovered that grid classes, especially the monotone ones, play a fundamental role in the understanding of the structure of general permutation classes. This motivates us to study the hardness of C-Pattern PPM for a (monotone) grid class C.We provide a complexity dichotomy for C-Pattern PPM when C is taken to be a monotone grid class. Specifically, we show that the problem is polynomial-time solvable if a certain graph associated with C, called the cell graph, is a forest, and it is NP-complete otherwise. We further generalize our results to grid classes whose blocks belong to classes of bounded grid-width. We show that the C-Pattern PPM for such a grid class C is polynomial-time solvable if the cell graph of C avoids a cycle or a certain special type of path, and it is NP-complete otherwise. ACM Subject ClassificationMathematics of computing → Permutations and combinations; Theory of computation → Pattern matching; Theory of computation → Problems, reductions and completeness Keywords and phrases permutations, pattern matching, grid classes

show abstract

Automatic discovery of structural rules of permutation classes

Cited by 4 publications

References 30 publications

On permutation patterns with constrained gap sizes

On permutation patterns with constrained gap sizes

Combinatorial Exploration: An algorithmic framework for enumeration

A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes

Contact Info

Product

Resources

About