1,029 research outputs found
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
Random Sorting Networks
A sorting network is a shortest path from 12...n to n...21 in the Cayley
graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform
random sorting network, as n->infinity the space-time process of swaps
converges to the product of semicircle law and Lebesgue measure. We conjecture
that the trajectories of individual particles converge to random sine curves,
while the permutation matrix at half-time converges to the projected surface
measure of the 2-sphere. We prove that, in the limit, the trajectories are
Holder-1/2 continuous, while the support of the permutation matrix lies within
a certain octagon. A key tool is a connection with random Young tableaux.Comment: 38 pages, 12 figure
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