Speaker: Prof. James Sellers (Penn State University, U.S.A.)
Title: Congruences for Fishburn Numbers
Time and Location: Wednesday, March 11, 2015
2:00 pm, Seminar room pond, RISC, Hagenberg
Abstract:
The Fishburn numbers, originally considered by Peter C. Fishburn, have been
shown to enumerate a variety of combinatorial objects. These include unlabelled
interval orders on n elements, (2+2)--avoiding posets with n elements, upper
triangular matrices with nonnegative integer entries and without zero rows or
columns such that the sum of all entries equals n, non--neighbor--nesting
matches on [2n], a certain set of permutations of [n] which serves as a natural
superset of the set of 231--avoiding permutations of [n], and ascent sequences
of length n.
In December 2013, Rob Rhoades (Stanford) gave a talk at Penn State in which he
described, among other things, the relationship between Fishburn numbers,
quantum modular forms, and Ramanujan's mock theta functions. Motivated by
Rhoades' talk, George Andrews and I were led to study the Fishburn numbers from
an arithmetic point of view - something which had not been done prior. In the
process, we proved that the Fishburn numbers satisfy infinitely many
Ramanujan--like congruences modulo certain primes p. In this talk, we will
describe this result in more detail as well as discuss how our work has served
as the motivation for a great deal of related work in the last year by Garvan,
Straub, and others.`