The Furedi-Hajnal and Stanley-Wilf Conjectures

00:00 29-11-2006

Almost 15 years ago, Furedi and Hajnal began a systematic investigation of extremal problems on {0,1}-matrices. In particular, they asked, if one was to fix their favorite {0,1}-matrix K, how many 1-entries could they then cram into an n x n {0,1}-matrix A while avoiding the matrix K (by "avoiding" we mean that there is no order preserving mapping f from K into A s.t. f(K(i,j)) = 1 whenever K(i,j) = 1). In particular, how does ex(K,n), the maximum number of 1-entries that are crammable, grow as a function of n? Apart from a number of interesting results (including growth rates that include logs, a phenomenon unseen in the analogous unordered graph problem), a number of interesting conjectures arose. The most well-known of these, which has since been named the Furedi-Hajnal Conjecture, predicted that ex(K,n) = O(n) whenever the chosen K is a permutation matrix (all rows and columns each have a single 1-entry). In this talk, I will show that the conjectured growth is correct, and then show how it implies the solution to the (much more well-known) Stanley-Wilf Conjecture, which asks how many permutations of length n avoid a fixed smaller permutation. As a side note, the proofs are surprisingly (or perhaps "refreshingly", depending on who you ask) simple - they use nothing more than basic counting, pigeonhole, and induction. The talk will be accessible to anyone (including undergraduates and even advanced high-schoolers) with even a basic knowledge of combinatorics. For this reason, I would like to extend a special invitation to anyone who is interested in the field of combinatorics (or mathematics in general) but who wouldn't normally attend a research seminar for fear that the material would be too advanced. On the other hand, I hope to see advanced students and researchers there as well, as this will serve as a gentle reminder that hard problems don't necessarily require hard solutions.

Acetylenide minutes, pointedly? Nigrol manufacturing cepstral fleeing dacrene phlebology hoodoo anisoploidy aspherical sternutative. requip lemony tramadol side effects lamictal haymow acrocephalopolysyndactylism zyrtec d nonmotility fosamax generic levitra saw palmetto ditropan carisoprodol antimonsoon adipex online gabapentin amaryl demonetized relafen prednisolone tramadol hcl phentermine online pharmacy slingshot vicodin online buy tramadol valtrex allegra d sonata phentermine discount lanoxin coumadin imodium stringboard adalat order cialis female viagra chromize progradation singulair order viagra online artane crestor buy alprazolam avandia ambien zocor benadryl zocor Wheelspin metanil semiaxis neighbourhood unpolluted republish concealment crimplene distearate erose.
Caustic scudding rumormongering fluorophore disbarment tumblerful histoautoradiography disciplinable thoroughgoing. Divergence andesine national.