685 research outputs found
Permutation classes
This is a survey on permutation classes for the upcoming book Handbook of
Enumerative Combinatorics
A decorated tree approach to random permutations in substitution-closed classes
We establish a novel bijective encoding that represents permutations as
forests of decorated (or enriched) trees. This allows us to prove local
convergence of uniform random permutations from substitution-closed classes
satisfying a criticality constraint. It also enables us to reprove and
strengthen permuton limits for these classes in a new way, that uses a
semi-local version of Aldous' skeleton decomposition for size-constrained
Galton--Watson trees.Comment: New version including referee's corrections, accepted for publication
in Electronic Journal of Probabilit
On The Möbius Function Of Permutations Under The Pattern Containment Order
We study several aspects of the Möbius function, μ[σ, π], on the poset of permutations under the pattern containment order.
First, we consider cases where the lower bound of the poset is indecomposable. We show that μ[σ, π] can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the Möbius function that only involves evaluating simple inequalities.
We then consider conditions on an interval which guarantee that the value of the Möbius function is zero. In particular, we show that if a permutation π contains two intervals of length 2, which are not order-isomorphic to one another, then μ[1, π] = 0. This allows us to prove that the proportion of permutations of length n with principal Möbius function equal to zero is asymptotically bounded below by (1−1/e) 2 ≥ 0.3995. This is the first result determining the value of μ[1, π] for an asymptotically positive proportion of permutations π.
Following this, we use “2413-balloon” permutations to show that the growth of the principal Möbius function on the permutation poset is exponential. This improves on previous work, which has shown that the growth is at least polynomial.
We then generalise 2413-balloon permutations, and find a recursion for the value of the principal Möbius function of these generalisations.
Finally, we look back at the results found, and discuss ways to relate the results from each chapter. We then consider further research avenues
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 monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
2-stack pushall sortable permutations
In the 60's, Knuth introduced stack-sorting and serial compositions of
stacks. In particular, one significant question arise out of the work of Knuth:
how to decide efficiently if a given permutation is sortable with 2 stacks in
series? Whether this problem is polynomial or NP-complete is still unanswered
yet. In this article we introduce 2-stack pushall permutations which form a
subclass of 2-stack sortable permutations and show that these two classes are
closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a
given permutation of size n is 2-stack pushall sortable and describe all its
sortings. This result is a step to the solve the general 2-stack sorting
problem in polynomial time.Comment: 41 page
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