473 research outputs found
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
Some open problems on permutation patterns
This is a brief survey of some open problems on permutation patterns, with an
emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns
in Permutations and words}. I first survey recent developments on the
enumeration and asymptotics of the pattern 1324, the last pattern of length 4
whose asymptotic growth is unknown, and related issues such as upper bounds for
the number of avoiders of any pattern of length for any given . Other
subjects treated are the M\"obius function, topological properties and other
algebraic aspects of the poset of permutations, ordered by containment, and
also the study of growth rates of permutation classes, which are containment
closed subsets of this poset.Comment: 20 pages. Related to upcoming talk at the British Combinatorial
Conference 2013. To appear in London Mathematical Society Lecture Note Serie
Recommended from our members
2413-balloon permutations and the growth of the Möbius function
We show that the growth of the principal Möbius function on the permutation poset is at least exponential. This improves on previous work, which has shown that the growth is at least polynomial.
We define a method of constructing a permutation from a smaller permutation which we call “ballooning”. We show that if β is a 2413-balloon, and π is the 2413-balloon of β, then μ[1, π] = 2μ[1, β]. This allows us to construct a sequence of per-mutations π1, π2, π3 ... with lengths n, n+4, n+8, ... such that μ[1, πi+1] = 2μ[1, πi], and this gives us exponential growth. Further, our construction method gives per-mutations that lie within a hereditary class with finitely many simple permutations.
We also find an expression for the value of μ[1, π], where π is a 2413-balloon, with no restriction on the permutation being ballooned
Word Measures on Symmetric Groups
Fix a word in a free group on generators. A -random
permutation in the symmetric group is obtained by sampling
independent uniformly random permutations and evaluating . In
[arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of
fixed points in a -random permutation is
, where is
the smallest rank of a subgroup containing as a non-primitive
element. We show that plays a role in estimates of all
"natural" families of characters of symmetric groups: those corresponding to
"stable" representations. In particular, we show that for all , the
average number of -cycles is
. As an application, we prove
that for every , every and every large enough , Schreier
graphs with random generators depicting the action of on
-tuples, have second eigenvalue at most
asymptotically almost surely. An important ingredient in this work is a
systematic study of not-necessarily connected Stallings core graphs.Comment: 50 pages, 2 figures. Extended abstract accepted to FPSAC 2020. Added
new Appendix A. Improved introductio
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