Monday January 11, 2010, 4:00 in Kemeny 004
Sergi Elizalde (Dartmouth College)
The number of numerical semigroups of a given genus
A numerical semigroup is a subset of the non-negative integers that contains 0, is closed under addition, and has a finite complement. The size of the complement is called the genus of the semigroup. In this talk I will give some lower and upper bounds on the number ng of numerical semigroups of genus g. The idea is to use a generating tree for all numerical semigroups, approximate it by other generating trees whose succession rules are easier to describe, and then obtain generating functions that enumerate their nodes. It is conjectured that the limit of ng+1/ng as g goes to infinity is the Golden ratio.
Monday January 18, 2010, 3:30 in Kemeny 004
Olivier Bernardi (MIT)
A unified bijective method for maps
A planar map is an embedding of connected planar graph in the sphere considered up to deformation. In the last decades, a bijective approach was developed by Schaeffer and others for studying maps. In this talk, I will present a general bijection between certain oriented maps and certain decorated plane trees. By specializing this bijection to particular classes of maps, we recover and extend several known bijections for triangulations, quadrangulations and bipartite
maps.
This is a joint work with Guillaume Chapuy and Eric Fusy.
Monday January 25, 2010, 4:00 in Kemeny 004
Rosa Orellana (Dartmouth College)
Descents and peaks
In this talk will present a combinatorial introduction to the descent and peak algebras. During the second part of the talk I will discuss recent work with Mathas on a generalization of descent algebras to complex reflection groups.
Thursday February 4, 2010, 1:00 in Kemeny 004
Paul Raff (Rutgers University)
Avoiding prescribed differences, and its generalizations
In this talk, I will primarily discuss the quantity
fΔ(n), defined as the size of the largest subset of [n]
avoiding differences prescribed in the set Δ. This can be
viewed in numerous other ways, including the size of the largest
independent set of circulant graphs. The impetus for investigation was
the short-lived Triangle Conjecture of Schutzenberger and Perrin and
its counterexample by Shor. A generalized recurrence equation will be
shown, leading to various consequences, including details on the
structure of the sequence {fΔ(n)} for any fΔ(n) and an
automated theorem prover (which will be demonstrated). The
investigations into fΔ(n) lead back to the Triangle Conjecture
in the form of an asymptotic proof. Additionally, three
generalizations will be briefly discussed, derived from extending the
idea of “difference”.
Monday February 8, 2010, 4:00 in Kemeny 004
Robert Brignall (University of Bristol)
Antichains and the structure of permutation classes
Permutation classes, the analogue of hereditary properties of graphs for
permutations, are defined as downsets in the permutation containment
partial ordering, and are most commonly described as the collection
avoiding some set of permutations, cf forbidden induced subgraphs for
hereditary graph properties. Much of the emphasis in their recent
intensive study has been on exact and asymptotic enumeration of particular
families of classes, but an ongoing study of the general structure of
permutations is yielding remarkable results which typically also have
significant enumerative consequences.
In this talk I will describe a number of these structural results useful
in studying the question of partial well-order — i.e. the existence or
otherwise of infinite antichains in a given permutation class. The
building blocks of all permutations are “simple permutations”, and we will
see how these on their own contribute to the partial well-order problem.
Seemingly independently, we will see how “grid classes”, a construction
used to express large complicated classes in terms of smaller
easily-described ones, also plays its part. Finally, I will present recent
and ongoing work in combining these two concepts, first describing a
general construction for antichains using “grid pin sequences”, and second
proving where certain families of grid classes are partially well-ordered.
Monday February 15, 2010
Seminar canceled
Monday February 22, 2010, 4:00 in Kemeny 004
John Schmitt (Middlebury College)
tba
Monday March 1, 2010, 4:00 in Kemeny 004
Matthew Kahle (Stanford University)
tba
Monday March 8, 2010, 4:00 in Kemeny 004
Jo Ellis-Monaghan (Saint Michael's College)
tba
Previous Terms:
Fall 2004.
Spring 2006.
Fall 2008
Winter 2009
Spring 2009
If you are interested in speaking please email Rosa Orellana, Sergi Elizalde, or Vince Vatter.