Structure of random 312‐avoiding permutations

  title={Structure of random 312‐avoiding permutations},
  author={Neal Madras and Lerna Pehlivan},
  journal={Random Structures \& Algorithms},
We evaluate the probabilities of various events under the uniform distribution on the set of 312‐avoiding permutations of 1,…,N . We derive exact formulas for the probability that the ith element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the… 
Limit Shapes of Restricted Permutations
Following the techniques initiated in \cite{MP}, we continue to study the limit shapes of random permutations avoiding a specific subset of patterns. We consider patterns in $S_3$ extensively, and
Patterns in random permutations avoiding the pattern 321
  • S. Janson
  • Mathematics
    Random Struct. Algorithms
  • 2019
A random permutation drawn from the set of 321-avoiding permutations of length n is considered and it is shown that the number of occurrences of another pattern sigma has a limit distribution.
Patterns in Random Permutations Avoiding the Pattern 132
  • S. Janson
  • Mathematics
    Combinatorics, Probability and Computing
  • 2016
A random permutation drawn from the set of 132-avoiding permutations of length n is considered and it is shown that the number of occurrences of another pattern σ has a limit distribution, after scaling by n λ(σ)/2.
Pattern-avoiding permutations and Brownian excursion, part II: fixed points
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we
Permutations with fixed pattern densities
The limit shapes of random permutations constrained by having fixed densities of a finite number of patterns are shown to be determined by maximizing entropy over permutons with those constraints.
Universal limits of substitution-closed permutation classes
We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider
On random combinatorial structures: partitions, permutations and asymptotic normality
Random combinatorial structures form an active field of research at the interface between combinatorics and probability theory. From a theoretical point of view, some of the main objectives are to
Pattern‐avoiding permutations and Brownian excursion part I: Shapes and fluctuations
The scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion are studied to strengthen the recent results and understand many of the interesting phenomena that had previously gone unexplained.


The shape of random pattern-avoiding permutations
Permutations with Restricted Patterns and Dyck Paths
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  • M. Bóna
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    J. Comb. Theory, Ser. A
  • 1997
Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating functionH(x) of all 1342-avoiding permutations of lengthnas well as
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