116 research outputs found
Some real and unreal enumerative geometry for flag manifolds
We present a general method for constructing real solutions to some problems
in enumerative geometry which gives lower bounds on the maximum number of real
solutions. We apply this method to show that two new classes of enumerative
geometric problems on flag manifolds may have all their solutions be real and
modify this method to show that another class may have no real solutions, which
is a new phenomenon. This method originated in a numerical homotopy
continuation algorithm adapted to the special Schubert calculus on
Grassmannians and in principle gives optimal numerical homotopy algorithms for
finding explicit solutions to these other enumerative problems.Comment: 19 pages, LaTeX-2e; Updated and final version. To appear in the issue
of Michigan Mathematical Journal dedicated to Bill Fulto
Cardinality of Rauzy classes
Rauzy classes define a partition of the set of irreducible (or
indecomposable) permutations. They were defined by G. Rauzy as part of an
induction algorithm for interval exchange transformations. In this article we
prove an explicit formula for the cardinality of all Rauzy classes.Comment: 43 pages, 22 figure
Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration
We consider the enumeration of pattern-avoiding involutions, focusing in
particular on sets defined by avoiding a single pattern of length 4. As we
demonstrate, the numerical data for these problems demonstrates some surprising
behavior. This strange behavior even provides some very unexpected data related
to the number of 1324-avoiding permutations
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