Abstract:  In "combinatorics" we start with some set of objects, like paths on a grid of length n, or permutations of length n, or graphs with n edges or vertices - with some (natural) restrictions - and we count the number of such things for each n. Then we look at the sequence of integers we get, see how they relate to others, and try to find compact formulas or descriptions for them. For example, count the number of arrangements of n left brackets and n right brackets, so that the brackets are "balanced". So, for n=2 we get 2 arrangements, (()) and ()(). For n=3 we get: ((())), ()()(), (())(), ()(()), (()()); a total of 5 arrangements.

In the past couple of years there has been a lot of interest in counting sets of permutations with some restrictions that I will describe in the talk (they avoid some patterns). Two years ago Marcus and Tardos resolved a longstanding conjecture of Stanley and Wilf, and last year Albert, me, Rechnitzer, Westcott and Zabrocki knocked over a conjecture of Arratia. I will try to explain these new theorems.

My most recent work involves permutations that are generated by passing an ordered sequence through some series of stacks. Knuth looked at this for one stack and came up with a nice answer. More than one stack and things get interesting.

No specific background on the part of the audience will be assumed. Students are encouraged to attend.