637 research outputs found
Intervals of Permutations with a Fixed Number of Descents are Shellable
The set of all permutations, ordered by pattern containment, is a poset. We
present an order isomorphism from the poset of permutations with a fixed number
of descents to a certain poset of words with subword order. We use this
bijection to show that intervals of permutations with a fixed number of
descents are shellable, and we present a formula for the M\"obius function of
these intervals. We present an alternative proof for a result on the M\"obius
function of intervals such that has exactly one descent. We
prove that if has exactly one descent and avoids 456123 and 356124, then
the intervals have no nontrivial disconnected subintervals; we
conjecture that these intervals are shellable
Asymptotic behavior of some statistics in Ewens random permutations
The purpose of this article is to present a general method to find limiting
laws for some renormalized statistics on random permutations. The model
considered here is Ewens sampling model, which generalizes uniform random
permutations. We describe the asymptotic behavior of a large family of
statistics, including the number of occurrences of any given dashed pattern.
Our approach is based on the method of moments and relies on the following
intuition: two events involving the images of different integers are almost
independent.Comment: 32 pages: final version for EJP, produced by the author. An extended
abstract of 12 pages, published in the proceedings of AofA 2012, is also
available as version
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