16 research outputs found
The absence of a pattern and the occurrences of another
Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern q in permutations that avoid another given pattern r. In some cases, we find the pattern that occurs least often, (resp. most often) in all r-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising
Prolific Compositions
Under what circumstances might every extension of a combinatorial structure
contain more copies of another one than the original did? This property, which
we call prolificity, holds universally in some cases (e.g., finite linear
orders) and only trivially in others (e.g., permutations). Integer
compositions, or equivalently layered permutations, provide a middle ground. In
that setting, there are prolific compositions for a given pattern if and only
if that pattern begins and ends with 1. For each pattern, there is an easily
constructed automaton that recognises prolific compositions for that pattern.
Some instances where there is a unique minimal prolific composition for a
pattern are classified
A relation on 132-avoiding permutation patterns
Rudolph conjectures that for permutations and of the same length,
for all if and only if the spine structure of is
less than or equal to the spine structure of in refinement order. We
prove one direction of this conjecture, by showing that if the spine structure
of is less than or equal to the spine structure of , then for all . We disprove the opposite direction by giving a
counterexample, and hence disprove the conjecture
On the Asymptotic Statistics of the Number of Occurrences of Multiple Permutation Patterns
We study statistical properties of the random variables ,
the number of occurrences of the pattern in the permutation . We
present two contrasting approaches to this problem: traditional probability
theory and the ``less traditional'' computational approach. Through the
perspective of the first one, we prove that for any pair of patterns
and , the random variables and are jointly
asymptotically normal (when the permutation is chosen from ). From the
other perspective, we develop algorithms that can show asymptotic normality and
joint asymptotic normality (up to a point) and derive explicit formulas for
quite a few moments and mixed moments empirically, yet rigorously. The
computational approach can also be extended to the case where permutations are
drawn from a set of pattern avoiders to produce many empirical moments and
mixed moments. This data suggests that some random variables are not
asymptotically normal in this setting.Comment: 18 page
Pattern Count on Multiply Restricted Permutations
Previous work has studied the pattern count on singly restricted
permutations. In this work, we focus on patterns of length 3 in multiply
restricted permutations, especially for double and triple pattern-avoiding
permutations. We derive explicit formulae or generating functions for various
occurrences of length 3 patterns on multiply restricted permutations, as well
as some combinatorial interpretations for non-trivial pattern relationships.Comment: 23 pages, 2 figure
Patterns in random permutations avoiding the pattern 132
We consider a random permutation drawn from the set of 132-avoiding
permutations of length and show that the number of occurrences of another
pattern has a limit distribution, after scaling by
where is the length of plus
the number of descents. The limit is not normal, and can be expressed as a
functional of a Brownian excursion. Moments can be found by recursion.Comment: 32 page
Equipopularity Classes of 132-Avoiding Permutations
The popularity of a pattern p in a set of permutations is the sum of the number of copies of p in each permutation of the set. We study pattern popularity in the set of 132-avoiding permutations. Two patterns are equipopular if, for all n, they have the same popularity in the set of length-n 132-avoiding permutations. There is a well-known bijection between 132-avoiding permutations and binary plane trees. The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines. Rudolph shows that patterns of the same length are equipopular if their associated binary plane trees have the same spine structure. We prove the converse of this result using the method of generating functions, which gives a complete classification of 132-avoiding permutations into equipopularity classes.Massachusetts Institute of Technology. Department of Mathematic