On dots in boxes, or Permutation pattern classes and regular languages
Abstract
This thesis investigates permutation pattern classes in a language theoretic context. Specifically
we explored the regularity of sets of permutations under the rank encoding. We found that the
subsets of plus- and minus-(in)decomposable permutations of a regular pattern class under the
rank encoding are also regular languages under that encoding. Further we investigated the sets of
permutations, which in their block-decomposition have the same simple permutation, and again
we found that these sets of permutations are regular languages under the rank encoding. This
natural progression from plus- and minus-decomposable to simple decomposable permutations led
us further to the set of simple permutations under the rank encoding, which we have also shown
to be regular under the rank encoding. This regular language enables us to find the set of simple
permutations of any class, independent of whether the class is regular under the rank encoding.
Furthermore the regularity of the languages of some types of classes is discussed. Under the
rank encoding we show that in general the skew-sum of classes, separable classes and wreath classes
are not regular languages; but that the direct-sum of classes, and with some restrictions on the
cardinality of the input classes the skew-sum and wreath sum of classes in fact are regular under
this encoding.
Other encodings such as the insertion encoding and the geometric grid encoding are discussed
and in the case of the geometric grid encoding alternative and constructive ways of retrieving the
basis of a geometric grid class are suggested.
The aforementioned results of the rank encoding have been implemented, amongst other previously
shown results, and tested. The program is available and accessible to everyone. We show
that the implementation for finding the block-decomposition of a permutation has cubic time complexity
with respect to the length of the permutation. The code for constructing the automaton
that accepts the language of all plus-indecomposable permutations of a regular class under the
rank encoding has quadratic time complexity with respect to the alphabet of the language. The
procedure to find the automaton that accepts the language of minus-decomposable permutations
has complexity O(k⁵) and we show that the implementation of the automaton to find the language
of simple permutations under the rank encoding has time complexity O(k⁵ 2ᵏ), where k is the size
of the alphabet. Further we show benchmark testing on previous important results involving the
rank encoding on classes and their bases.
Type
Thesis, PhD Doctor of Philosophy
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