25,595 research outputs found
Lattice congruences, fans and Hopf algebras
We give a unified explanation of the geometric and algebraic properties of
two well-known maps, one from permutations to triangulations, and another from
permutations to subsets. Furthermore we give a broad generalization of the
maps. Specifically, for any lattice congruence of the weak order on a Coxeter
group we construct a complete fan of convex cones with strong properties
relative to the corresponding lattice quotient of the weak order. We show that
if a family of lattice congruences on the symmetric groups satisfies certain
compatibility conditions then the family defines a sub Hopf algebra of the
Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has
a basis which is described by a type of pattern-avoidance. Applying these
results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite
sequence of smaller algebras, where the second algebra in the sequence is the
Hopf algebra of non-commutative symmetric functions. We also associate both a
fan and a Hopf algebra to a set of permutations which appears to be
equinumerous with the Baxter permutations.Comment: 34 pages, 1 figur
Sylow -groups of polynomial permutations on the integers mod
We describe the Sylow -groups of the group of polynomial permutations of
the integers mod
Lower bounds for Kazhdan-Lusztig polynomials from patterns
We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in
a Weyl group W in terms of "patterns''. This is expressed by a "pattern map"
from W to W' for any parabloic subgroup W'. This notion generalizes the concept
of patterns and pattern avoidance for permutations to all Weyl groups. The main
tool of the proof is a "hyperbolic localization" on intersection cohomology;
see the related paper http://front.math.ucdavis.edu/math.AG/0202251Comment: 14 pages; updated references. Final version; will appear in
Transformation Groups vol.8, no.
Combinatorial specification of permutation classes
This article presents a methodology that automatically derives a
combinatorial specification for the permutation class C = Av(B), given its
basis B of excluded patterns and the set of simple permutations in C, when
these sets are both finite. This is achieved considering both pattern avoidance
and pattern containment constraints in permutations.The obtained specification
yields a system of equations satisfied by the generating function of C, this
system being always positiveand algebraic. It also yields a uniform random
sampler of permutations in C. The method presentedis fully algorithmic
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