Speaker: John D’Angelo (UIUC)
Title: Combinatorial arrays motivated by bracket tournaments
Abstract: We begin by introducing an array that results from studying equivalence classes of tournament brackets.
We establish some basic results and describe an open problem about asymptotics. One partitions each row sum into three pieces, and wishes to know the percentage of each piece in the limit. For the bracket problem the limits are approximately .7440, .2236, .0223, but I do not know a closed formula for them. We then observe that everything we did can be significantly generalized, in fact every monotone (non-decreasing) sequence of positive integers yields an array with similar properties. In the bracket case, the powers of two serve as the input sequence. Using the positive integers as the input sequence yields the well-known Catalan triangle. The asymptotic result is elementary for the Catalan triangle: the three limits are 1/2, 5/16, 3/16.
Using the Fibonacci numbers (or the primes) leads to something that seems to be completely new.
