16,271 research outputs found
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
A Spectral Approach to Consecutive Pattern-Avoiding Permutations
We consider the problem of enumerating permutations in the symmetric group on
elements which avoid a given set of consecutive pattern , and in
particular computing asymptotics as tends to infinity. We develop a general
method which solves this enumeration problem using the spectral theory of
integral operators on , where the patterns in has length
. Kre\u{\i}n and Rutman's generalization of the Perron--Frobenius theory
of non-negative matrices plays a central role. Our methods give detailed
asymptotic expansions and allow for explicit computation of leading terms in
many cases. As a corollary to our results, we settle a conjecture of Warlimont
on asymptotics for the number of permutations avoiding a consecutive pattern.Comment: a reference is added; corrected typos; to appear in Journal of
Combinatoric
The Brownian limit of separable permutations
We study random uniform permutations in an important class of
pattern-avoiding permutations: the separable permutations. We describe the
asymptotics of the number of occurrences of any fixed given pattern in such a
random permutation in terms of the Brownian excursion. In the recent
terminology of permutons, our work can be interpreted as the convergence of
uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion
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
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