Combinatorics Seminar

Spring 2009

The seminar is usually on Thursdays.

The seminar will be after the Thursday Lunch Expedition.


Thursday May 7, 2009
Peter McNamara (Bucknell University)

Title:
  The Schur-Positivity Poset

Kemeny 201 and TIME: 1:30 PM

Abstract: Determining relations among symmetric functions continues to be a topic of considerable interest in algebraic combinatorics. We will focus on relations among Schur functions and their most classical generalization, the skew Schur functions. To any skew shape A, we can associate the skew Schur function s_A. We can then order the set of skew shapes by saying that A \leq B if s_B - s_A is Schur-positive, i.e., when expanded in the basis of Schur functions, all the coefficients are non-negative. We call the resulting poset the "Schur-positivity poset" on skew shapes. While much recent work on Schur-positivity can be formulated in terms of the Schur-positivity poset, a complete understanding of the poset is presently well out of reach. After giving the necessary background and introducing the Schur-positivity poset, we will present _necessary_ conditions on the shapes of A and B for A \leq B. We will conclude with broad open questions in the area.


Thursday April 30, 2009
Alexey Spiridonov (MIT)

Title:
  Pattern-avoidance in binary fillings of grid shapes

Kemeny 201 and TIME: 1:30 PM

Abstract: A "grid shape" is a set of boxes chosen from a square grid; any Young diagram is an example. We consider a notion of pattern-avoidance for 0-1 fillings of grid shapes, which generalizes permutation pattern-avoidance. A filling avoids some patterns if none of its sub-shapes equal any of the patterns. We focus on patterns that are pairs of 2 x 2 fillings. For some shapes, fillings that avoid specic 2 x 2 pairs are in bijection with totally nonnegative Grassmann cells, or with acyclic orientations of bipartite graphs. We prove a number of results analogous to Wilf-equivalence for these objects --- that is, we show that in various classes of shapes, certain pattern pairs are equally restrictive. As an application of our results, we are able to give *all* the equivalences, which hold for every Young shape (and some other classes). In this talk, I will also give an overview of a bijection that proves one of the equivalences.


Thursday April 23, 2009
Lou Shapiro (Howard University)

Title:
  Trees, Logs, and the Riordan Group

Kemeny 201 and TIME: 1:30 PM

Abstract: We will discuss the V = TL equation for classes ordered trees, use it to give some easy proofs of some recent results and see how the exponential version of the Riordan group come into the picture. Among the classes considered are ordered trees, 0.1.2 (Motzkin) trees even trees complete and incomplete binary trees, even trees, and red-green trees. The methods employed end up being row reduction, matrix multiplication, generating functions, and a semidirect product.




Previous Terms:

Fall 2004.

Spring 2006.

Fall 2008

Winter 2009

If you are interested in speaking please email Rosa.C.Orellana "at" Dartmouth "dot" edu or Sergi Elizalde

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